Key Concept: Differentiation, Chain Rule

a) $e^{2x} (2\sin(3x) + 3\cos(3x))$

[Solution Description]

To find $\frac{dy}{dx}$, we use the product rule and chain rule. Let $u = e^{2x}$ and $v = \sin(3x)$. Then:

$\frac{dy}{dx} = u' \cdot v + u \cdot v'$

First, compute $u'$:

$u' = \frac{d}{dx} e^{2x} = e^{2x} \cdot 2 = 2e^{2x}$

Next, compute $v'$:

$v' = \frac{d}{dx} \sin(3x) = \cos(3x) \cdot 3 = 3\cos(3x)$

Now, substitute into the product rule:

$\frac{dy}{dx} = 2e^{2x} \cdot \sin(3x) + e^{2x} \cdot 3\cos(3x) = e^{2x} (2\sin(3x) + 3\cos(3x))$

Key Concept: Introduction

c) To present background information

[Solution Description] The purpose of an introduction in academic writing is to provide an overview of the topic, present the main objectives or thesis, and outline the structure of the work. It aims to engage the reader and give them a clear idea of what to expect in the subsequent sections.

Key Concept: Integration, Substitution

b) $-\sqrt{1 - x^2} + C$

[Solution Description]

To evaluate the integral $\int \frac{x}{\sqrt{1 - x^2}} \, dx$, we use substitution. Let $u = 1 - x^2$. Then, $du = -2x \, dx$, or equivalently, $-\frac{1}{2}\, du = x\, dx$.

Substitute into the integral:

$\int \frac{x}{\sqrt{1 - x^2}} \, dx = \int \frac{1}{\sqrt{u}} \cdot -\frac{1}{2} \, du = -\frac{1}{2} \int u^{-1/2} \, du$

Now, integrate $u^{-1/2}$:

$-\frac{1}{2} \int u^{-1/2}\, du = -\frac{1}{2} \cdot \frac{u^{1/2}}{1/2} + C = -\sqrt{u} + C$

Finally, substitute back for $u$:

$-\sqrt{1 - x^2} + C$

Key Concept: Definition of Exponents

b) 8

[Solution Description]

The expression $2^3$ represents 2 multiplied by itself 3 times. So, $2^3 = 2 \times 2 \times 2 = 8$.

Key Concept: Exponents, Division, Large Numbers

a) $1.145 \times 10^{-2}$

[Solution Description] To find the fraction representing the mass of the satellite compared to the planet, we divide the mass of the satellite by the mass of the planet. Given: Mass of the planet = $6.42 \times 10^{14}$ kg and Mass of the satellite = $7.35 \times 10^{12}$ kg. We use the formula: $\frac{7.35 \times 10^{12}}{6.42 \times 10^{14}}$. First, divide the coefficients: $\frac{7.35}{6.42} \approx 1.145$. Then, subtract the exponents: $12 - 14 = -2$. So, the result is $1.145 \times 10^{-2}$. Therefore, the mass of the satellite is $1.145 \times 10^{-2}$ times the mass of the planet.

Key Concept: Definition of Exponents

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

The assertion states that $2^{-2} = \frac{1}{4}$. According to the definition of exponents, a negative exponent indicates the reciprocal of the base raised to the positive exponent. Therefore, $2^{-2} = \frac{1}{2^2} = \frac{1}{4}$. Both the assertion and the reason are true, and the reason correctly explains the assertion.

Key Concept: Exponents, Large numbers

c) $3.011 \times 10^{24}$

[Solution Description]

To find the number of atoms in 5 grams of hydrogen, multiply the number of atoms in 1 gram by 5.

$6.022 \times 10^{23} \times 5 = 30.11 \times 10^{23}$

This can be written as $3.011 \times 10^{24}$.

Key Concept: Writing large numbers using exponents

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

First, let's understand the assertion. The number $10^{6}$ is calculated as follows:

$10^{6} = 10 \times_{1} 10 \times_{2} 10 \times_{3} 10 \times_{4} 10 \times_{5} 10$

Performing the multiplication step by step:

$10 \times_{1} 10 = 100$

$100 \times_{2} 10 = 1,000$

$1,000 \times_{3} 10 = 10,000$

$10,000 \times_{4} 10 = 100,000$

$100,000 \times_{5} 10 = 1,000,000$

So, $10^{6} = 1,000,000$, which confirms that the assertion is true.

Now, let's examine the reason. The reason states that in the expression $10^{n}$, when n is a positive integer, it represents the product of multiplying 10 by itself n times. This is a fundamental property of exponents with positive integers, and it directly explains why $10^{6}$ equals 1,000,000. Therefore, the reason is true and correctly explains the assertion.

Key Concept: Exponents, Large numbers

b) $3.978 \times 10^{23}$

[Solution Description]

To find the number of spaceships required to match the mass of the sun, divide the mass of the sun by the mass of one spaceship.

$\frac{1.989 \times 10^{30}}{5 \times 10^6} = 0.3978 \times 10^{24} = 3.978 \times 10^{23}$

Key Concept: Understanding Powers

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

The assertion states that $2^{-2}$ is equal to $\frac{1}{4}$. This is correct because, by definition, a negative exponent indicates the reciprocal of the base raised to the positive exponent. Therefore, $2^{-2} = \frac{1}{2^{2}} = \frac{1}{4}$. The reason correctly explains why the assertion is true.

Key Concept: Understanding Powers

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description] The assertion states that $2^{3} = 8$. This is true because $2^{3}$ means multiplying 2 by itself three times: $2 \times 2 \times 2 = 8$. The reason correctly explains why the assertion is true by describing the process of calculating $2^{3}$. Therefore, both the assertion and reason are true, and the reason is the correct explanation of the assertion.

Key Concept: Understanding Negative Exponents

c) $\frac{1}{4}$

[Solution Description] The value of $2^{-2}$ can be calculated using the rule for negative exponents, which states that $a^{-n} = \frac{1}{a^n}$. Applying this rule to $2^{-2}$, we get $2^{-2} = \frac{1}{2^2}$. Now, calculate $2^2$ which is 4. Therefore, $2^{-2} = \frac{1}{4}$.

Key Concept: Base and exponent notation

b) $1$

[Solution Description]

First, simplify the numerator:

$3^4 \times 3^{-6} = 3^{4-6} = 3^{-2}$

Now, divide by the denominator:

$\frac{3^{-2}}{3^{-2}} = 3^{-2-(-2)} = 3^{0} = 1$

Therefore, the simplified expression is 1.

Key Concept: Base and Exponent Notation

b) 2

[Solution Description]

In the expression $3^2$, the number 3 is the base and 2 is the exponent. The exponent indicates how many times the base is multiplied by itself. Therefore, the exponent is 2.

Key Concept: Base and Exponent Notation

b) 8

[Solution Description]

The expression $2^3$ means 2 multiplied by itself 3 times. So, $2^3 = 2 \times 2 \times 2$. Calculating this: $2 \times 2 = 4$, and $4 \times 2 = 8$. Therefore, the value of $2^3$ is 8.

Key Concept: Expanding exponents

b) $x^6$

[Solution Description]

To expand $(x^2)^3$, we use the power of a power rule, which states that $(a^m)^n = a^{m \cdot n}$. Applying this rule, $(x^2)^3 = x^{2 \cdot 3} = x^6$.

Key Concept: Expanding expressions with exponents

b) $7 \times 10^{1} + 0 \times 10^{0} + 8 \times 10^{-1} + 0 \times 10^{-2} + 9 \times 10^{-3}$

[Solution Description]

To expand 70.809 using exponents, we break it down as follows:

The number 70.809 can be written as:

$7 \times 10^{1} + 0 \times 10^{0} + 8 \times 10^{-1} + 0 \times 10^{-2} + 9 \times 10^{-3}$

Here is the step-by-step breakdown:

1. The digit 7 is in the tens place, so it is multiplied by $10^{1}$.

2. The digit 0 is in the ones place, so it is multiplied by $10^{0}$.

3. The digit 8 is in the tenths place, so it is multiplied by $10^{-1}$.

4. The digit 0 is in the hundredths place, so it is multiplied by $10^{-2}$.

5. The digit 9 is in the thousandths place, so it is multiplied by $10^{-3}$.

Thus, the expanded form of 70.809 using exponents is:

$7 \times 10^{1} + 0 \times 10^{0} + 8 \times 10^{-1} + 0 \times 10^{-2} + 9 \times 10^{-3}$

Key Concept: Expanding exponents

b) $z^{12}$

[Solution Description]

To expand $(z^4)^3$, we use the power of a power rule, which states that $(a^m)^n = a^{m \cdot n}$. Applying this rule, $(z^4)^3 = z^{4 \cdot 3} = z^{12}$.

Key Concept: Exponent Rules, Product of Exponents

b) 6

[Solution Description]

Simplify the expression step by step:

$(2^3 \times 3^2) \div (2^2 \times 3^1) = \frac{2^3 \times 3^2}{2^2 \times 3^1}$

Use the properties of exponents to simplify:

$\frac{2^3}{2^2} = 2^{3-2} = 2^1 = 2$

$\frac{3^2}{3^1} = 3^{2-1} = 3^1 = 3$

Multiply the simplified terms:

$2 \times 3 = 6$

Therefore, the simplified expression is equal to 6.

Key Concept: Exponential Notation, Multiplication of Exponents

c) $3.575 \times 10^{51}$

[Solution Description]

To find the number of hydrogen atoms that make up the mass of the Earth, divide the mass of the Earth by the mass of a single hydrogen atom:

$\text{Number of atoms} = \frac{\text{Mass of Earth}}{\text{Mass of one hydrogen atom}} = \frac{5.97 \times 10^{24}}{1.67 \times 10^{-27}}$

Simplify the division by separating the coefficients and the exponents:

$\text{Number of atoms} = \frac{5.97}{1.67} \times 10^{24 - (-27)} = \frac{5.97}{1.67} \times 10^{51}$

Calculate the division of the coefficients:

$\frac{5.97}{1.67} \approx 3.575$

Therefore, the number of hydrogen atoms is approximately $3.575 \times 10^{51}$.

Key Concept: Negative Exponents, Exponential Notation

b) $\frac{8}{9}$

[Solution Description]

First, calculate the value of $x = 3^{-2}$. Using the property of negative exponents, $a^{-n} = \frac{1}{a^n}$, we get:

$x = 3^{-2} = \frac{1}{3^2} = \frac{1}{9}$

Next, calculate the value of $y = 2^{-3}$:

$y = 2^{-3} = \frac{1}{2^3} = \frac{1}{8}$

Now, compute $\frac{x}{y}$:

$\frac{x}{y} = \frac{\frac{1}{9}}{\frac{1}{8}} = \frac{1}{9} \times \frac{8}{1} = \frac{8}{9}$

Therefore, the value of $\frac{x}{y}$ is $\frac{8}{9}$.

Key Concept: Simplifying large and small numbers

c) Assertion is true, but Reason is false.

[Solution Description]

The number $0.000007$ m has the decimal point moved 6 places to the right to become $7$, which gives an exponent of $-6$. Therefore, the standard form is $7 \times 10^{-6}$ m. The reason states that in standard form, the exponent is equal to the number of places the decimal point is moved to the right, which is correct for positive exponents but not for negative exponents. For negative exponents, the number of places the decimal point is moved to the right corresponds to the absolute value of the exponent. Thus, the assertion is true, but the reason is false.

Key Concept: Simplifying large and small numbers, standard form

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

To add masses in scientific notation, we need to ensure that the exponents are the same. The mass of the Earth is $5.97 \times 10^{24} kg$ and the mass of the Moon is $7.35 \times 10^{22} kg$. We can convert the mass of the Moon to match the exponent of the Earth's mass by multiplying it by 100: $7.35 \times 10^{22} kg = 0.0735 \times 10^{24} kg$. Now we can add the two masses: $5.97 \times 10^{24} kg + 0.0735 \times 10^{24} kg = 6.0435 \times 10^{24} kg$. Rounding this to one significant figure gives us $6 \times 10^{24} kg$, which matches the assertion. The reason correctly explains why the exponents need to be the same before addition.

Key Concept: Standard Form of Small Numbers

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description] To express $0.000005$ in standard form, we need to write it as a number between 1 and 10 multiplied by a power of 10. First, we move the decimal point to the right until we get a number between 1 and 10. Here, moving the decimal point 6 places to the right gives us $5$. Since we moved the decimal point to the right, the exponent of 10 will be negative. Therefore, $0.000005 = 5 \times 10^{-6}$. The Reason correctly explains that in standard form, small numbers are represented as a decimal number between 1 and 10 multiplied by a power of 10. Thus, both Assertion and Reason are true, and the Reason is the correct explanation for the Assertion.

Key Concept: Powers with Negative Exponents

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

Both the Assertion and Reason are true. The Assertion correctly states that $a^{-n} = \frac{1}{a^n}$ for any non-zero integer $a$. The Reason correctly explains that this is due to the reciprocal nature of negative exponents. Therefore, both statements are true, and the Reason is the correct explanation of the Assertion.

Key Concept: Negative Exponents, Simplification

b) $\frac{1}{9}$

[Solution Description]

First, simplify $3^{-7} \div 3^{-10}$. Using the rule $\frac{a^m}{a^n} = a^{m-n}$, we get:

$3^{-7} \div 3^{-10} = 3^{-7 - (-10)} = 3^{3}$

Now multiply the result by $3^{-5}$:

$3^{3} \times 3^{-5} = 3^{3 + (-5)} = 3^{-2}$

Finally, express $3^{-2}$ in positive exponent form using the rule $a^{-n} = \frac{1}{a^n}$:

$3^{-2} = \frac{1}{3^2}$

So the final answer is $\frac{1}{9}$.

Key Concept: Powers with Negative Exponents

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

The assertion states that $3^{-2} = \frac{1}{9}$. Let us verify this using the reason provided. According to the rule of negative exponents, $a^{-n} = \frac{1}{a^n}$ for any non-zero integer $a$. Applying this to $3^{-2}$, we get $3^{-2} = \frac{1}{3^2} = \frac{1}{9}$. Thus, both the assertion and the reason are true, and the reason correctly explains the assertion.

Key Concept: Powers with Negative Exponents

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

First, let's understand the Assertion. The expression $10^{-10}$ can be rewritten using the property of negative exponents, which states that $a^{-n} = \frac{1}{a^n}$. Applying this property, we get:

$10^{-10} = \frac{1}{10^{10}}$

This confirms that the Assertion is true.

Now, let's analyze the Reason. The Reason states that as the exponent decreases by 1, the value becomes one-tenth of the previous value. This is consistent with the pattern observed in the syllabus:

$10^{0} = 1, \quad 10^{-1} = \frac{1}{10}, \quad 10^{-2} = \frac{1}{100}, \quad 10^{-3} = \frac{1}{1000}, \quad \text{and so on.}$

Each time the exponent decreases by 1, the value becomes one-tenth of the previous value. Therefore, the Reason is also true.

Furthermore, the Reason correctly explains why the Assertion holds true. The pattern described in the Reason is precisely what leads to the equality $10^{-10} = \frac{1}{10^{10}}$. Hence, the Reason is the correct explanation of the Assertion.

Key Concept: Negative Exponents

b) $\frac{1}{8}$

[Solution Description] To find the value of $2^{-3}$, we use the property of negative exponents, which states that $a^{-n} = \frac{1}{a^n}$. Applying this, $2^{-3} = \frac{1}{2^3}$. Calculating $2^3$ gives $2 \times 2 \times 2 = 8$. Therefore, $2^{-3} = \frac{1}{8}$.

Key Concept: Negative Exponents

d) 4

[Solution Description]

Given that $2^{-x} = \frac{1}{16}$, we can rewrite $\frac{1}{16}$ as $2^{-4}$ because $2^4 = 16$. Therefore:

$2^{-x} = 2^{-4} \implies -x = -4 \implies x = 4$

Hence, the value of $x$ is 4.

Key Concept: Powers with Negative Exponents, Multiplicative Inverse Using Exponents

b) $\frac{1}{1125}$

[Solution Description]

First, simplify the expression $(3^{–2} \times 5^{–3})^{–1}$.

Using the property $a^{–m} = \frac{1}{a^m}$, we get:

$3^{–2} = \frac{1}{3^2} = \frac{1}{9}$

$5^{–3} = \frac{1}{5^3} = \frac{1}{125}$

Now multiply these two results:

$\frac{1}{9} \times \frac{1}{125} = \frac{1}{1125}$

Apply the exponent $^{–1}$ to the product:

$\left(\frac{1}{1125}\right)^{–1} = 1125$

The multiplicative inverse of 1125 is $\frac{1}{1125}$.

Key Concept: Multiplicative Inverse Using Exponents

a) 125

[Solution Description]

The multiplicative inverse of $a^{-m}$ is $a^m$. Here, $a = 5$ and $m = 3$. So, the multiplicative inverse of $5^{-3}$ is $5^3$.

Key Concept: Multiplicative Inverse, Negative Exponents

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

The assertion states that if $a = 5$, then the multiplicative inverse of $a^{-3}$ is $a^3$. Let's verify this:

By definition, the multiplicative inverse of a number is such that when multiplied by the original number, it gives 1. So, the multiplicative inverse of $a^{-3}$ should satisfy:

$a^{-3} \times (\text{multiplicative inverse}) = 1$

Substituting $a = 5$:

$5^{-3} \times (\text{multiplicative inverse}) = 1$

We know that $5^{-3} = \frac{1}{5^3} = \frac{1}{125}$. So, the multiplicative inverse would be $5^3 = 125$. Thus, the assertion is true.

The reason states that for any non-zero integer $a$, $a^{-m}$ is the multiplicative inverse of $a^m$. This is correct as per the definition of negative exponents and multiplicative inverses. Hence, the reason is also true and correctly explains the assertion.

Key Concept: Expanding Numbers Using Negative Exponents

b) $3 \times 10^2 + 0 \times 10^1 + 7 \times 10^0 + 8 \times 10^{-1} + 2 \times 10^{-2}$

[Solution Description]

To expand 307.82 using negative exponents, we break it down as follows:

$307.82 = 3 \times 10^2 + 0 \times 10^1 + 7 \times 10^0 + 8 \times 10^{-1} + 2 \times 10^{-2}$

Here, $3 \times 10^2$ represents the hundreds place, $0 \times 10^1$ represents the tens place, $7 \times 10^0$ represents the ones place, $8 \times 10^{-1}$ represents the tenths place, and $2 \times 10^{-2}$ represents the hundredths place.

Key Concept: Negative Exponents, Expanded Form

a) $3 \times 10^3 + 0 \times 10^2 + 4 \times 10^1 + 0 \times 10^0 + 8 \times 10^{-1} + 1 \times 10^{-2} + 7 \times 10^{-3}$

[Solution Description]

To expand 3040.817 using exponents, we break it down into its place values:

$3040.817 = 3 \times 10^3 + 0 \times 10^2 + 4 \times 10^1 + 0 \times 10^0 + 8 \times 10^{-1} + 1 \times 10^{-2} + 7 \times 10^{-3}$

Thus, the expanded form is $3 \times 10^3 + 0 \times 10^2 + 4 \times 10^1 + 0 \times 10^0 + 8 \times 10^{-1} + 1 \times 10^{-2} + 7 \times 10^{-3}$.

Key Concept: Expanding Numbers Using Negative Exponents

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

The assertion states that the number 1025.63 can be expanded using positive and negative exponents. To verify this, we break down the number into its place values:

$1025.63 = 1 \times 1000 + 0 \times 100 + 2 \times 10 + 5 \times 1 + \frac{6}{10} + \frac{3}{100}$

Converting these into exponential form:

$1 \times 10^3 + 0 \times 10^2 + 2 \times 10^1 + 5 \times 10^0 + 6 \times 10^{-1} + 3 \times 10^{-2}$

The assertion is correct. The reason explains that negative exponents are used to represent fractional parts of numbers in expanded form, which is also true. Moreover, the reason correctly explains the assertion.

Key Concept: Negative Exponents

a) $10^{-5}$

[Solution Description] To find the exponential form of $0.00001$, we first express it as a fraction: $0.00001 = \frac{1}{100000}$. Since $100000$ is equal to $10^5$, the fraction becomes $\frac{1}{10^5}$. Applying the rule of negative exponents, $\frac{1}{10^5} = 10^{-5}$. Hence, the exponential form of $0.00001$ is $10^{-5}$.

Key Concept: Multiplicative Inverse

b) 16

[Solution Description]

The multiplicative inverse of $a^{-m}$ is $a^m$. Thus, the multiplicative inverse of $2^{-4}$ is calculated as follows:

$2^{-4} = \frac{1}{2^4} = \frac{1}{16}$

Therefore, the multiplicative inverse of $2^{-4}$ is $2^4 = 16$.

Key Concept: Negative Exponents

c) $\frac{1}{1000}$

[Solution Description]

The formula for negative exponents is $a^{-m} = \frac{1}{a^m}$. Using this, we can calculate $10^{-3}$ as follows:

$10^{-3} = \frac{1}{10^3} = \frac{1}{1000}$

Therefore, the value of $10^{-3}$ is $\frac{1}{1000}$.

Key Concept: Laws of Exponents

b) 1/15625[Solution Description]To simplify $(5^{2})^{-3}$, we use the law of exponents which states that $(a^m)^n = a^{m n}$. Here, $a = 5$, $m = 2$, and $n = -3$. Therefore, $(5^{2})^{-3} = 5^{2 \times -3} = 5^{-6}$. Now, calculate $5^{-6} = \frac{1}{5^6} = \frac{1}{15625}$.

Key Concept: Exponent Subtraction, Zero Exponent

b) 1

[Solution Description]

To simplify $5^3 \times 5^{-5} \times 5^2$, we use the law of exponents that states $a^m \times a^n = a^{m+n}$. Here, $a = 5$, $m = 3$, $n = -5$, and $p = 2$. Applying the law:

$5^3 \times 5^{-5} \times 5^2 = 5^{3 + (-5) + 2} = 5^{0}$

Using the zero exponent rule $a^0 = 1$:

$5^{0} = 1$

The simplified form of the expression is 1.

Key Concept: Laws of Exponents

d) 4

[Solution Description]

To simplify $\frac{2^{-3}}{2^{-5}}$, we use the law of exponents which states that $\frac{a^m}{a^n} = a^{m - n}$. Here, $a = 2$, $m = -3$, and $n = -5$. Therefore, $\frac{2^{-3}}{2^{-5}} = 2^{-3 - (-5)} = 2^{2}$. Now, calculate $2^{2} = 2 \times 2 = 4$.

Key Concept: Product Law of Exponents

c) $2^2$

[Solution Description]

To simplify the expression $2^{-3} \times 2^5$, we use the product law of exponents which states that for any non-zero integer $a$, $a^m \times a^n = a^{m + n}$. Applying this law, we get:

$2^{-3} \times 2^5 = 2^{-3 + 5} = 2^2$

Therefore, the simplified form of the expression is $2^2$.

Key Concept: Laws of Exponents: Product Law

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

The assertion states that the product law $a^m \times a^n = a^{m + n}$ holds true for any non-zero integer $a$ and integers $m$ and $n$. This is correct as demonstrated by examples in the syllabus, such as $2^{-3} \times 2^{-2} = 2^{-5}$, $(-3)^{-4} \times (-3)^{-3} = (-3)^{-7}$, and $5^{-2} \times 5^4 = 5^2$. The reason provided explains that this consistency is due to the fundamental property of exponents. Since the assertion is true and the reason correctly explains why the assertion is true, both are valid.

Key Concept: Product Law

b) $z^{15}$

[Solution Description]

Apply the product law of exponents by adding the exponents since the base is the same. Thus, $z^6 \cdot z^9 = z^{6+9} = z^{15}$.

Key Concept: Quotient Law, Positive Exponents

c) $7^4$

[Solution Description]

To simplify $(7^{-3} \div 7^{-5}) \times 7^2$, first apply the Quotient Law of Exponents to the division part:

$7^{-3} \div 7^{-5} = 7^{-3 - (-5)} = 7^{2}$

Now multiply this result by $7^2$:

$7^{2} \times 7^2 = 7^{2 + 2} = 7^4$

Key Concept: Quotient Law, Negative Exponents

b) $\frac{9}{25}$

[Solution Description]

To simplify $\left(\frac{5}{3}\right)^{-4} \div \left(\frac{5}{3}\right)^{-2}$, we use the Quotient Law of Exponents:

$\left(\frac{5}{3}\right)^{-4} \div \left(\frac{5}{3}\right)^{-2} = \left(\frac{5}{3}\right)^{-4 - (-2)} = \left(\frac{5}{3}\right)^{-2}$

Now, to express this with a positive exponent, we take the reciprocal:

$\left(\frac{5}{3}\right)^{-2} = \left(\frac{3}{5}\right)^2 = \frac{9}{25}$

Key Concept: Quotient Law

b) 49

[Solution Description]

Using the Quotient Law of Exponents, we know that $\frac{a^m}{a^n} = a^{m-n}$. Here, $a = 7$, $m = 5$, and $n = 3$. Applying the law, we get:

$\frac{7^5}{7^3} = 7^{5-3} = 7^2$

Calculating $7^2$:

$7^2 = 49$

Therefore, the simplified form of the expression is $49$.

Key Concept: Power of a Power Law, Laws of Exponents

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

Let us analyze the Assertion and Reason step by step.

**Assertion Analysis**:

Given that $a$ is a non-zero integer, and $m$ and $n$ are integers, we need to verify whether $(a^{-m})^n = a^{-mn}$.

According to the Power of a Power Law, $(a^m)^n = a^{mn}$. It does not matter if the exponents are positive or negative; the law still holds. Therefore:

$(a^{-m})^n = a^{(-m) \times n} = a^{-mn}$

So, the Assertion is true.

**Reason Analysis**:

The Reason states that the Power of a Power Law is given by $(a^m)^n = a^{mn}$ where $m$ and $n$ are integers. This statement is correct and directly supports the Assertion.

**Conclusion**:

Both the Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Power of a Power Law

b) $5^6$

[Solution Description]

To simplify $(5^3)^2$, we use the Power of a Power Law, which states that $(a^m)^n = a^{mn}$. Here, $a = 5$, $m = 3$, and $n = 2$. Applying the law gives $(5^3)^2 = 5^{3 \times 2} = 5^6$.

Key Concept: Power of a Power Law, Negative Exponents

c) $4$

[Solution Description]

Given the equation $(5^{-3})^x = 5^{-12}$, use the power of a power law, $(a^m)^n = a^{mn}$. Applying the law, we get $(5^{-3})^x = 5^{-3x}$. Now, set this equal to $5^{-12}$: $5^{-3x} = 5^{-12}$. Since the bases are the same, the exponents must be equal: $-3x = -12$. Solve for $x$ by dividing both sides by $-3$: $x = \frac{-12}{-3} = 4$. Therefore, the value of $x$ is $4$.

Key Concept: Product to Power Law

b) 64

[Solution Description]

The Product to Power Law states that $(a^m)^n = a^{mn}$. Applying this law, we get:

$(2^3)^2 = 2^{3 \times 2} = 2^6 = 64$

Key Concept: Product to Power Law

a) $\frac{1}{4}$

[Solution Description]

To simplify the expression $(-2)^{-5} \times (-2)^3$, we use the Product to Power Law: $a^m \times a^n = a^{m + n}$. Here, $a = -2$, $m = -5$, and $n = 3$. Applying the law, we get:

$(-2)^{-5} \times (-2)^3 = (-2)^{-5 + 3} = (-2)^{-2}$

Now, $(-2)^{-2} = \frac{1}{(-2)^2} = \frac{1}{4}$, so the simplified form is $\frac{1}{4}$.

Key Concept: Product to Power Law

a) $-10$

[Solution Description]

Given the expression $x^{-4} \times x^{-6}$, we use the Product to Power Law: $a^m \times a^n = a^{m + n}$. Here, $a = x$, $m = -4$, and $n = -6$. Applying the law, we get:

$x^{-4} \times x^{-6} = x^{-4 + (-6)} = x^{-10}$

Therefore, the value of $k$ is $-10$.

Key Concept: Quotient to Power Law

a) $\frac{m^{10}}{n^6}$

[Solution Description]

To simplify $(\frac{m^5}{n^3})^2$, apply the Quotient to Power Law, which states $(\frac{x}{y})^n = \frac{x^n}{y^n}$:

$(\frac{m^5}{n^3})^2 = \frac{(m^5)^2}{(n^3)^2}$

Next, use the Power of a Power Law, $(x^m)^n = x^{m \cdot n}$, to simplify the exponents:

$\frac{(m^5)^2}{(n^3)^2} = \frac{m^{5 \cdot 2}}{n^{3 \cdot 2}} = \frac{m^{10}}{n^6}$

The simplified form is $\frac{m^{10}}{n^6}$.

Key Concept: Quotient to Power Law

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

The Quotient to Power Law is a fundamental property in exponents, which states that $\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}$ for non-zero integers $a$ and $b$, and integer $m$. This law is directly applied to simplify expressions involving exponents of fractions. Here, the assertion correctly applies this law to simplify $\left(\frac{a}{b}\right)^m$ to $\frac{a^m}{b^m}$, and the reason accurately states the Quotient to Power Law. Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Quotient to Power Law

c) $\frac{16}{81}$

[Solution Description]

The Quotient to Power Law states that for any non-zero integers a and b, and any integer m, $\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}$. Here, a=2, b=3 and m=4.

So, $\left(\frac{2}{3}\right)^4 = \frac{2^4}{3^4} = \frac{16}{81}$

Key Concept: Zero Exponent Rule, Even Exponents

b) 2

[Solution Description]

Given that $x$ is a non-zero integer and $k$ is an even integer, we know that $x^k = 1$. According to the Zero Exponent Rule, $x^0 = 1$. Therefore, $x^0 + x^k = 1 + 1 = 2$.

Key Concept: Zero Exponent Rule

c) 1

[Solution Description]

According to the zero exponent rule, any non-zero number raised to the power of 0 equals 1. Since -3 is a non-zero number, $(-3)^0 = 1$.

Key Concept: Zero Exponent Rule

c) 1

[Solution Description]

According to the Zero Exponent Rule, any non-zero number raised to the power of 0 is equal to 1. Here, the base is $-7$, which is a non-zero number. Therefore, $(-7)^0 = 1$.

Key Concept: Standard Form

b) $1.275 \times 10^{-5}$

[Solution Description]

To express 0.00001275 in standard form, we need to write it as a number between 1 and 10 multiplied by a power of 10.

First, move the decimal point to the right until the number becomes between 1 and 10:

$0.00001275 \rightarrow 1.275$

Count how many places the decimal moved: it moved 5 places to the right.

Since the decimal was moved to the right, the exponent will be negative:

$0.00001275 = 1.275 \times 10^{-5}$

Key Concept: Use of Exponents to Express Small Numbers in Standard Form

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

The assertion states that 0.000035 can be expressed as $3.5 \times 10^{-5}$ in standard form. To verify this, we move the decimal point five places to the right to get 3.5, and since we moved the decimal point to the right, the exponent is negative. Thus, 0.000035 is correctly expressed as $3.5 \times 10^{-5}$. The reason explains that in standard form, a number is written as $a \times 10^{n}$, where $1 \leq a < 10$ and $n$ is an integer. This definition aligns with the process used in the assertion. Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Standard Form, Subtraction of Exponents

b) $1.49216 \times 10^{11}$

[Solution Description]

To find the distance between the Sun and Moon, subtract the distance between Earth and Moon from the distance between Sun and Earth:

$1.496 \times 10^{11} - 3.84 \times 10^8 = (1.496 \times 1000 \times 10^8) - (3.84 \times 10^8) = (1496 - 3.84) \times 10^8 = 1492.16 \times 10^8$

Expressing $1492.16 \times 10^8$ in standard form:

$1.49216 \times 10^{11}$

Key Concept: Addition in Standard Form

a) $6.0435 \times 10^{24}$ kg

[Solution Description]

To add the masses of the Earth and the Moon, we first express them with the same exponent:

$5.97 \times 10^{24} = 597 \times 10^{22}$

Now, add the two masses:

$597 \times 10^{22} + 7.35 \times 10^{22} = (597 + 7.35) \times 10^{22} = 604.35 \times 10^{22}$

To express this in standard form, we adjust the coefficient so that it is between 1 and 10:

$604.35 \times 10^{22} = 6.0435 \times 10^{24}$

Therefore, the total mass of the Earth and the Moon is $6.0435 \times 10^{24}$ kg.

Key Concept: Scientific Notation

b) $7 \times 10^{-6}$ mm

[Solution Description]

To express $0.000007$ mm in standard form, we move the decimal point to the right until it is after the first non-zero digit (7). This requires moving the decimal point 6 places to the right. Therefore, $0.000007 = 7 \times 10^{-6}$ mm.

Key Concept: Standard Form Expression

a) $4.56 \times 10^{-5}$

[Solution Description]

To express 0.0000456 in standard form, we need to write it as $a \times 10^{n}$, where $1 \leq a < 10$ and $n$ is an integer.

First, move the decimal point until it is after the first non-zero digit:

$0.0000456 = 4.56 \times 10^{-5}$

Therefore, the standard form of 0.0000456 is $4.56 \times 10^{-5}$.

Key Concept: Comparing Large Numbers

c) 100 times

[Solution Description]

To find how many times larger the Sun's diameter is compared to the Earth's diameter, we divide the two values:

$\frac{1.4 \times 10^{9}}{1.2756 \times 10^{7}} = \frac{1.4}{1.2756} \times 10^{9-7} = 1.097 \times 10^{2} \approx 100$

So, the diameter of the Sun is approximately 100 times larger than the diameter of the Earth.

Key Concept: Standard Form, Usual Form

b) $0.00001275$ m

[Solution Description]

To convert $1.275 \times 10^{-5}$ m into usual form, we need to move the decimal point 5 places to the left because the exponent is -5. This gives us $0.00001275$ m. Therefore, the usual form is $0.00001275$ m.

Key Concept: Expressing Large Numbers in Standard Form

b) $1.496 \times 10^{11}$ m

[Solution Description]

To express 149,600,000,000 m in standard form, we first identify the coefficient by moving the decimal point to the left until we have a number between 1 and 10. In this case, we get 1.496. Next, we count how many places we moved the decimal point, which is 11 places to the left. Therefore, the standard form of 149,600,000,000 m is $1.496 \times 10^{11}$ m.

Key Concept: Assertion and Reason

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description] To convert the distance from Earth to the Sun into standard form, we start with the given number 149,600,000,000 meters. We move the decimal point to the left until we have a number between 1 and 10, which gives us 1.496. We count the number of places we moved the decimal point, which is 11 places. Therefore, the distance from the Earth to the Sun in standard form is $1.496 \times 10^{11}$ meters. The reason states that standard form is used to express very large or very small numbers in a compact form, which is true and explains why we use standard form for this purpose.

Key Concept: Standard Form Representation

c) $1.496 \times 10^{11}$

[Solution Description]

To express 149,600,000,000 in standard form, we move the decimal point to the left until there is only one non-zero digit before the decimal point. We count how many places we moved the decimal point, which is 11 places. Thus, the standard form is $1.496 \times 10^{11}$.

Key Concept: Use of Exponents to Express Small Numbers in Standard Form

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

The distance from the Earth to the Sun is given as $149,600,000,000$ m. To express this in standard form, we move the decimal point to the left until there is only one non-zero digit to the left of the decimal. This requires moving the decimal point 11 places, resulting in $1.496 \times 10^{11}$ m. Therefore, both the Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Exponent Rules, Conversion

b) $2.8$

[Solution Description]

The diameter of the red blood cell is $7 \times 10^{-6}$ meters and the thickness is $2.5 \times 10^{-6}$ meters. To find the ratio of the diameter to the thickness, divide the diameter by the thickness:

$\frac{7 \times 10^{-6}}{2.5 \times 10^{-6}} = \frac{7}{2.5} = 2.8$

Therefore, the ratio of the diameter to the thickness is $2.8$.

Key Concept: Exponent Representation

c) $7 \times 10^{-6}$ m

[Solution Description]

In exponential form, 0.000007 m can be written as $7 \times 10^{-6}$ m.

Key Concept: Standard Form, Scientific Notation

b) $2.5 \times 10^{-6}$

[Solution Description]

The thickness of the red blood cell is given as $0.0000025$ meters. To express this in scientific notation, we move the decimal point to the right until it is after the first non-zero digit (which is 2 here). We moved the decimal point 6 places to the right, so we multiply by $10^{-6}$. Therefore, the thickness in scientific notation is $2.5 \times 10^{-6}$ meters.

Key Concept: Adding Large Numbers

a) $604.35 \times 10^{22}$ kg

[Solution Description] To find the total mass of the Earth and the Moon, we add their masses. First, we convert both masses to the same power of 10 for ease of calculation: $5.97 \times 10^{24}$ kg = $597 \times 10^{22}$ kg. Now, we can add the masses: $597 \times 10^{22}$ + $7.35 \times 10^{22}$ = $(597 + 7.35) \times 10^{22}$ = $604.35 \times 10^{22}$ kg.

Key Concept: Use of Exponents to Express Small Numbers in Standard Form

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

The diameter of the Earth is given as $1.2756 \times 10^7$ m, and the diameter of the Sun is $1.4 \times 10^9$ m. To compare these two diameters, we divide the diameter of the Sun by the diameter of the Earth:

$\frac{1.4 \times 10^9}{1.2756 \times 10^7} = \frac{1.4 \times 10^{9-7}}{1.2756} = \frac{1.4 \times 100}{1.2756} \approx 100$

The calculation shows that the diameter of the Sun is approximately 100 times the diameter of the Earth. Therefore, both Assertion and Reason are true, and Reason correctly explains Assertion.

Key Concept: Comparing Large and Small Numbers

a) 9.34

[Solution Description]

To find how many times the mass of Earth is greater than the mass of Mars, we divide the mass of Earth by the mass of Mars.

$\frac{5.97 \times 10^{24}}{6.39 \times 10^{23}} = \frac{5.97}{6.39} \times 10^{24-23} = \frac{5.97}{6.39} \times 10^{1} = \frac{5.97}{6.39} \times 10$

$\frac{5.97}{6.39} = 0.934$ (approx.). Therefore, $\frac{5.97}{6.39} \times 10 = 9.34$. So, the mass of Earth is approximately 9.34 times the mass of Mars.

Key Concept: Comparing very large and very small numbers

c) 12,756,000

[Solution Description]

To express the diameter of the Earth in meters without scientific notation, we need to convert $1.2756 \times 10^7$ to a standard number.

$1.2756 \times 10^7$ means moving the decimal point 7 places to the right.

Starting with 1.2756, we move the decimal point 7 places to the right:

1.2756 → 12.756 → 127.56 → 1,275.6 → 12,756 → 127,560 → 1,275,600 → 12,756,000

So, the diameter of the Earth is 12,756,000 meters.

Key Concept: Comparing very large and very small numbers

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description] The assertion states that the diameter of the Sun is approximately 100 times the diameter of the Earth. To verify this, we compare the given diameters: Diameter of the Sun = $1.4 \times 10^9$ m and Diameter of the Earth = $1.2756 \times 10^7$ m. We calculate the ratio of the diameters as follows: $\frac{1.4 \times 10^9}{1.2756 \times 10^7} = \frac{1.4 \times 10^{9-7}}{1.2756} = \frac{1.4 \times 10^2}{1.2756} \approx 100$. This confirms that the assertion is true. The reason provides the diameters of the Sun and the Earth, which are used to calculate the ratio, thus correctly explaining the assertion.

Key Concept: Use of Exponents, Standard Form

a) $1.839 \times 10^{3}$

[Solution Description]

To find the ratio of the mass of a neutron to the mass of an electron, we divide the two masses:

$\frac{1.675 \times 10^{-27}}{9.109 \times 10^{-31}} = \frac{1.675}{9.109} \times 10^{-27 - (-31)} = \frac{1.675}{9.109} \times 10^{4}$

Calculating $\frac{1.675}{9.109} \approx 0.1839$, so the ratio is:

$0.1839 \times 10^{4} = 1.839 \times 10^{3}$

Key Concept: Addition and Subtraction in Standard Form

a) $5.5 \times 10^{-5}$

[Solution Description]

To add two numbers in standard form, ensure the exponents are the same. Here, both numbers have the same exponent of $10^{-5}$. Add the coefficients: 3.4 + 2.1 = 5.5. Thus, the result is $5.5 \times 10^{-5}$.

Key Concept: Standard Form

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

The distance between the Sun and Earth is given as $1.496 \times 10^{11} m$, and the distance between Earth and Moon is $3.84 \times 10^{8} m$. To find the distance between Sun and Moon, we subtract the two distances, but they must have the same exponent. Therefore, we convert $1.496 \times 10^{11} m$ to $1496 \times 10^{8} m$ and then perform the subtraction: $(1496 - 3.84) \times 10^{8} m = 1492.16 \times 10^{8} m$. Thus, both Assertion and Reason are true, and Reason correctly explains Assertion.

Key Concept: Standard Form, Subtraction

a) $8.45 \times 10^{-5}$

[Solution Description]

To subtract $7.5 \times 10^{-6}$ from $9.2 \times 10^{-5}$, first express both numbers with the same exponent. Convert $7.5 \times 10^{-6}$ to $0.75 \times 10^{-5}$. Now, subtract $0.75 \times 10^{-5}$ from $9.2 \times 10^{-5}$ to get $8.45 \times 10^{-5}$.

Key Concept: Converting exponents to the same power

b) $3.66 \times 10^{-5}$

[Solution Description]

To add the numbers, they must have the same exponent. Convert $2.6 \times 10^{-6}$ to $0.26 \times 10^{-5}$. Now, add the numbers: $3.4 \times 10^{-5} + 0.26 \times 10^{-5} = 3.66 \times 10^{-5}$.

Key Concept: Exponents

c) $7.2 \times 10^{-6}$

[Solution Description] To convert $0.0000072$ to standard form, we move the decimal point to the right until it is after the first non-zero digit. This gives $7.2$. We moved the decimal point 6 places to the right, so the exponent is $-6$. Therefore, the standard form is $7.2 \times 10^{-6}$.

Key Concept: Exponents

c) $9.1 \times 10^{-8}$

[Solution Description] To convert $0.000000091$ to standard form, we move the decimal point to the right until it is after the first non-zero digit. This gives $9.1$. We moved the decimal point 8 places to the right, so the exponent is $-8$. Therefore, the standard form is $9.1 \times 10^{-8}$.

Key Concept: Exponent Laws, Division of Exponents

a) $4^0$

[Solution Description]

First, simplify the numerator $(4^{-3} \times 4^5)$ using the law of exponents $a^m \times a^n = a^{m+n}$:

$4^{-3} \times 4^5 = 4^{-3 + 5} = 4^2$

Next, divide by the denominator $(4^2)$ using the law of exponents $\frac{a^m}{a^n} = a^{m-n}$:

$\frac{4^2}{4^2} = 4^{2 - 2} = 4^0 = 1$

The expression simplifies to 1, which can be expressed in exponential form as $4^0$.

Key Concept: Mass of Earth, Mass of Moon

c) $5.8965 \times 10^{24}$ kg

[Solution Description]

To find the difference between the masses, subtract the mass of Moon from the mass of Earth. First, express both masses with the same exponent:

Mass of Earth = $5.97 \times 10^{24}$ kg = $597 \times 10^{22}$ kg

Mass of Moon = $7.35 \times 10^{22}$ kg

Now, subtract the two masses:

Difference = $597 \times 10^{22} - 7.35 \times 10^{22}$ = $(597 - 7.35) \times 10^{22}$ = $589.65 \times 10^{22}$ kg

Convert this to standard form:

$589.65 \times 10^{22}$ kg = $5.8965 \times 10^{24}$ kg

Key Concept: Total Mass Calculation

a) $604.35 \times 10^{22}$ kg

[Solution Description]

To find the total mass, we add the mass of the Earth and the Moon. First, express both masses with the same exponent of 10:

$5.97 \times 10^{24} = 597 \times 10^{22}$

Now, add the two masses:

$597 \times 10^{22} + 7.35 \times 10^{22} = (597 + 7.35) \times 10^{22} = 604.35 \times 10^{22} \text{ kg}$

Therefore, the total mass is $604.35 \times 10^{22}$ kg.

Key Concept: Total mass calculation

a) $6.609 \times 10^{24}$ kg

[Solution Description]

Express both masses with the same exponent. Convert $5.97 \times 10^{24}$ kg to $59.7 \times 10^{23}$ kg by dividing 5.97 by 10. Now, add the two masses:

$59.7 \times 10^{23} + 6.39 \times 10^{23} = (59.7 + 6.39) \times 10^{23} = 66.09 \times 10^{23} \text{ kg}$

Key Concept: Total mass calculation

c) $6.0435 \times 10^{23}$ kg

[Solution Description]

To find the total mass, express both masses with the same exponent. Let's convert $5.97 \times 10^{24}$ kg to $597 \times 10^{22}$ kg by multiplying 5.97 by 100. Now, add the two masses:

$597 \times 10^{22} + 7.35 \times 10^{22} = (597 + 7.35) \times 10^{22} = 604.35 \times 10^{22} \text{ kg}$

Key Concept: Use of Exponents to Express Small Numbers in Standard Form

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

To express $0.00001275$ in standard form, we move the decimal point to the right until it is after the first non-zero digit (1). The number of places the decimal point is moved determines the exponent. Here, the decimal point is moved 5 places to the right, so the exponent is $-5$. Therefore, $0.00001275 = 1.275 \times 10^{-5}$. The assertion is true. The reason states that for a number less than 1, the exponent in standard form is negative, and its absolute value equals the number of places the decimal point is moved to the right. This is also true and correctly explains the assertion.

Key Concept: Standard Form Conversion

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

To express 0.000003 m in standard form, we move the decimal point to the right until it is after the first non-zero digit (which is 3). This requires moving the decimal point 6 places to the right, resulting in $3 \times 10^{-6}$ m. The reason correctly states that in standard form, a number is written as $a \times 10^n$, where $1 \leq |a| < 10$ and $n$ is an integer. Here, $a = 3$ and $n = -6$, which satisfies the condition. Therefore, both the assertion and reason are true, and the reason is the correct explanation of the assertion.

Key Concept: Standard Form - Multiplication and Division

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

To verify the assertion, let us multiply the numbers in standard form: $1.5 \times 10^{11}$ and $3 \times 10^{8}$.

Step 1: Multiply the coefficients: $1.5 \times 3 = 4.5$.

Step 2: Add the exponents: $11 + 8 = 19$.

Step 3: Combine the results to get the product in standard form: $4.5 \times 10^{19}$.

The assertion is true as the product matches the given result. The reason provided states that when multiplying numbers in standard form, we add the exponents, which is correct. Therefore, both assertion and reason are true, and reason correctly explains the assertion.

Key Concept: Standard Form, Division

a) $3 \times 10^{2}$

[Solution Description]

To find the quotient of $C$ and $D$, divide the coefficients and subtract the exponents:

$(6 \times 10^{-4}) \div (2 \times 10^{-6}) = (6 \div 2) \times 10^{-4 - (-6)} = 3 \times 10^{2}$.

The result is already in standard form.

Key Concept: Expressing Distance in Standard Form

d) $3.84 \times 10^{8}$ m

[Solution Description]

The given distance is already in standard form since $3.84$ is greater than or equal to 1 and less than 10, and the exponent $n$ is 8. Therefore, the standard form of the distance is $3.84 \times 10^{8}$ m.

Key Concept: Standard Form Conversion

a) $1.49216 \times 10^{11}$ m

[Solution Description]

To express $1492.16 \times 10^{8}$ m in standard form with an exponent of 11, we move the decimal three places to the left and adjust the exponent accordingly. Thus, $1492.16 \times 10^{8}$ m = $1.49216 \times 10^{11}$ m.

Key Concept: Distance between Sun and Moon

a) $1.49216 \times 10^{11}$ m

[Solution Description]

The distance between the Sun and Moon is calculated by subtracting the distance between Earth and Moon from the distance between Sun and Earth:

$\text{Distance between Sun and Moon} = 1.496 \times 10^{11} - 3.84 \times 10^{8}$

To simplify, we convert $1.496 \times 10^{11}$ to $1496 \times 10^{8}$:

$1.496 \times 10^{11} = 1496 \times 10^{8}$

Now, subtract $3.84 \times 10^{8}$ from $1496 \times 10^{8}$:

$1496 \times 10^{8} - 3.84 \times 10^{8} = (1496 - 3.84) \times 10^{8} = 1492.16 \times 10^{8} \text{ m}$

Therefore, the distance between the Sun and Moon is $1492.16 \times 10^{8}$ m.

Key Concept: Distance between Sun and Moon, Exponential Operations

c) 388.57

[Solution Description]

To find how many times larger the distance between the Sun and Moon is compared to the distance between the Earth and Moon, divide the two distances: $\frac{1.49216 \times 10^{11}}{3.84 \times 10^{8}}$. Simplify the division: $\frac{1.49216}{3.84} \times 10^{11-8} = 0.38857 \times 10^{3} = 388.57$.