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: 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: Introduction: Introduction

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

[Solution Description]

The assertion states that the concept of introduction is fundamental in understanding any subject. This is true because an introduction sets the stage for learning by providing context and basic principles. The reason claims that introduction provides a foundational overview, which is essential for building deeper knowledge. This is also true and directly supports the assertion. Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Exponents, Large Numbers

a) $3.33 \times 10^5$

[Solution Description] To find how many times greater the mass of the star is compared to the mass of the Earth, we divide the mass of the star by the mass of the Earth. Given: Mass of the star = $1.989 \times 10^{30}$ kg and Mass of the Earth = $5.97 \times 10^{24}$ kg. We use the formula: $\frac{1.989 \times 10^{30}}{5.97 \times 10^{24}}$. First, divide the coefficients: $\frac{1.989}{5.97} \approx 0.333$. Then, subtract the exponents: $30 - 24 = 6$. So, the result is $0.333 \times 10^6$. Since $0.333 \times 10^6$ can be written as $3.33 \times 10^5$, the mass of the star is $3.33 \times 10^5$ times greater than the mass of the Earth.

Key Concept: Definition of Exponents

d) 128

[Solution Description]

To solve $2^3 \times 2^4$, we use the property of exponents that states $a^m \times a^n = a^{m+n}$. Here, $a = 2$, $m = 3$, and $n = 4$. So, $2^3 \times 2^4 = 2^{3+4} = 2^7$. Now, calculate $2^7$ which is $2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 128$.

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 $2^{3}$ is equal to $8$. This is correct because $2^{3} = 2 \times 2 \times 2 = 8$. The reason given is that $2^{3}$ means $2$ multiplied by itself $3$ times, which is the correct definition of an exponent. Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Exponents, Large numbers

b) $4.5 \times 10^{11}$ meters

[Solution Description]

To calculate the distance traveled by light, multiply the speed of light by the time taken.

$3 \times 10^8 \times 1.5 \times 10^3 = 4.5 \times 10^{11} \text{ meters}$

Key Concept: Exponents

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

[Solution Description]

To find the value of $2^{-3}$, we use the property of exponents that states $a^{-n} = \frac{1}{a^n}$. Here, $a$ is 2 and $n$ is 3. So, substituting the values, we get:

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

Therefore, the answer is $\frac{1}{8}$.

Key Concept: Exponents

c) 1000

[Solution Description]

To divide two numbers with the same base, subtract the exponent in the denominator from the exponent in the numerator. So:

$\frac{10^{5}}{10^{2}} = 10^{5-2} = 10^{3}.$

Calculating $10^{3}$ gives us 1000. Therefore, the answer is 1000.

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 $2^{-3}$ equals $\frac{1}{8}$. Let us verify this using the reason provided. According to the rule of exponents, for any non-zero integer $a$ and positive integer $n$, $a^{-n} = \frac{1}{a^n}$. Here, $a = 2$ and $n = 3$. Applying the rule, $2^{-3} = \frac{1}{2^3}$. Calculating $2^3$, we get $2 \times 2 \times 2 = 8$. Therefore, $2^{-3} = \frac{1}{8}$. Both the assertion and reason are true, and the reason correctly explains the assertion.

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: Scientific Notation

b) $5.97 \times 10^{24}$ kg

[Solution Description] The mass of the earth is given as 5,970,000,000,000,000,000,000,000 kg. To write this in scientific notation, we move the decimal point to the left until there is only one non-zero digit to the left of the decimal point, resulting in 5.97. We count the number of places we moved the decimal point, which is 24. Therefore, the mass of the earth in scientific notation is $5.97 \times 10^{24}$ kg.

Key Concept: Base and exponent notation

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 $\frac{1}{4}$. To verify this, we can use the definition of a negative exponent, which states that $a^{-n} = \frac{1}{a^n}$. Applying this to $2^{-2}$, we get $2^{-2} = \frac{1}{2^2} = \frac{1}{4}$. Therefore, the assertion is true.

The reason given is that a negative exponent indicates the reciprocal of the base raised to the absolute value of the exponent. This is a correct definition of a negative exponent. Since both the assertion and the reason are true, and the reason correctly explains the assertion, the correct option is:

Key Concept: Base and exponent notation

a) $1.90 \times 10^{27}$ kg

[Solution Description]

To find the mass of Jupiter, we multiply the mass of Earth by 318. The mass of Earth is $5.97 \times 10^{24}$ kg. So, the mass of Jupiter is:

$5.97 \times 10^{24} \times 318 = 1.89846 \times 10^{27} \text{ kg}$

Therefore, the approximate mass of Jupiter in scientific notation is $1.90 \times 10^{27}$ kg.

Key Concept: Base and Exponent Notation

b) 5

[Solution Description]

In the expression $5^4$, the number 5 is the base and 4 is the exponent. The base is the number that is being raised to a power. Therefore, the base is 5.

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

a) $3 \times 10^3 + 1 \times 10^2 + 5 \times 10^1 + 0 \times 10^0 + 7 \times 10^{-1} + 2 \times 10^{-2}$

[Solution Description]

To write the expanded form of \$3150.72\$ using exponents, we decompose the number as follows:

$3150.72 = 3 \times 10^3 + 1 \times 10^2 + 5 \times 10^1 + 0 \times 10^0 + 7 \times 10^{-1} + 2 \times 10^{-2}$

Here, \$10^{-1} = \frac{1}{10}\$ and \$10^{-2} = \frac{1}{100}\$. Therefore, the correct expanded form is given by the equation above.

Key Concept: Expanding expressions with exponents

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

[Solution Description]

The assertion states that the number $1256.249$ can be expanded using exponents as $1 \times 10^3 + 2 \times 10^2 + 5 \times 10^1 + 6 \times 10^0 + 2 \times 10^{-1} + 4 \times 10^{-2} + 9 \times 10^{-3}$. This is correct because each digit is multiplied by a power of 10 corresponding to its position relative to the decimal point. The reason explains the concept of expanding numbers using exponents, which is also correct and directly supports the assertion. Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

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: Exponents

b) 1000000000000000000000000

[Solution Description]

The expression $10^{24}$ means 10 multiplied by itself 24 times, which is a very large number. It is equivalent to 1 followed by 24 zeros.

Key Concept: Importance of Exponents in Mathematics

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

[Solution Description]

The assertion states that exponents are crucial in simplifying large numbers and performing complex calculations efficiently. This is true because exponents provide a way to represent very large or very small numbers in a manageable form, such as scientific notation. The reason states that exponents allow us to express repeated multiplication in a compact form, which is essential for mathematical operations. This is also true because exponents are fundamentally used to simplify the representation of repeated multiplication (e.g., $2^3 = 2 \times 2 \times 2$). Moreover, the reason correctly explains why exponents are crucial, as their ability to compactly represent repeated multiplication directly contributes to their efficiency in calculations. Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

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

b) 0.00042

[Solution Description]

To convert $4.2 \times 10^{-4}$ to decimal form, move the decimal point 4 places to the left.

Starting with 4.2, moving the decimal point 4 places gives 0.00042.

Therefore, the decimal equivalent of the number is 0.00042.

Key Concept: Simplifying large and small numbers

c) 7500000

[Solution Description]

To convert $7.5 \times 10^6$ to standard form, move the decimal point 6 places to the right.

Starting with 7.5, moving the decimal point 6 places gives 7500000.

Therefore, the standard form of the number is 7500000.

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

d) 29

[Solution Description]

First, simplify each term using the rule $\left(\frac{a}{b}\right)^{-m} = \left(\frac{b}{a}\right)^m$. Thus:

$\left(\frac{1}{2}\right)^{-2} = 2^2 = 4,\quad \left(\frac{1}{3}\right)^{-2} = 3^2 = 9,\quad \left(\frac{1}{4}\right)^{-2} = 4^2 = 16$

Now add these results together:

$4 + 9 + 16 = 29$

So the final answer is 29.

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: Negative Exponents, Pattern Recognition

c) $\frac{1}{32}$

[Solution Description]

To find the value of $2^{-5}$, we can use the pattern of negative exponents. The negative exponent indicates the reciprocal of the base raised to the positive exponent. So, $2^{-5} = \frac{1}{2^5}$. Now, calculate $2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32$. Therefore, $2^{-5} = \frac{1}{32}$.

Key Concept: Negative Exponents

b) $\frac{1}{25}$

[Solution Description]

Using the formula for negative exponents, $a^{-n} = \frac{1}{a^n}$. For $a = 5$ and $n = 2$, we get:

$5^{-2} = \frac{1}{5^2} = \frac{1}{25}$

Therefore, $5^{-2} = \frac{1}{25}$.

Key Concept: Negative Exponents, Multi-step Solution

c) 4

[Solution Description]

Given that $3^{-n} = \frac{1}{81}$, we can rewrite this as $3^{-n} = \frac{1}{3^4}$ since $3^4 = 81$. This implies that $3^{-n} = 3^{-4}$. Therefore, $-n = -4$, and solving for $n$ gives $n = 4$.

Key Concept: Multiplicative Inverse

a) $125$

[Solution Description]

To find the multiplicative inverse of $5^{-3}$, we use the property $a^{-m} = \frac{1}{a^{m}}$. Here, $a = 5$ and $m = 3$. So, $5^{-3} = \frac{1}{5^{3}} = \frac{1}{125}$. The multiplicative inverse of $\frac{1}{125}$ is $125$ because $\frac{1}{125} \times 125 = 1$. Therefore, the multiplicative inverse of $5^{-3}$ is $125$.

Key Concept: Powers with Negative Exponents

d) Assertion is false, but Reason is true.

[Solution Description]

To find the multiplicative inverse of $5^{-3}$, we use the formula $a^{-m} = \frac{1}{a^m}$. Applying this to $5^{-3}$, we get $\frac{1}{5^3} = \frac{1}{125}$. Therefore, the multiplicative inverse of $5^{-3}$ is $\frac{1}{125}$. The assertion states that the multiplicative inverse is $125$, which is incorrect. However, the reason correctly states the general formula for negative exponents. Thus, the Assertion is false, but the Reason is true.

Key Concept: Multiplicative Inverse

a) $49$

[Solution Description]

Applying the property $a^{-m} = \frac{1}{a^{m}}$, where $a = 7$ and $m = 2$. So, $7^{-2} = \frac{1}{7^{2}} = \frac{1}{49}$. The multiplicative inverse of $\frac{1}{49}$ is $49$ because $\frac{1}{49} \times 49 = 1$. Therefore, the multiplicative inverse of $7^{-2}$ is $49$.

Key Concept: Expanding Numbers Using Negative Exponents

a) $1 \times 10^3 + 2 \times 10^2 + 5 \times 10^1 + 6 \times 10^0 + 2 \times 10^{-1} + 4 \times 10^{-2} + 9 \times 10^{-3}$

[Solution Description]

To expand 1256.249 using negative exponents, we break it down as follows:

$1256.249 = 1 \times 10^3 + 2 \times 10^2 + 5 \times 10^1 + 6 \times 10^0 + 2 \times 10^{-1} + 4 \times 10^{-2} + 9 \times 10^{-3}$

Here, $1 \times 10^3$ represents the thousands place, $2 \times 10^2$ represents the hundreds place, $5 \times 10^1$ represents the tens place, $6 \times 10^0$ represents the ones place, $2 \times 10^{-1}$ represents the tenths place, $4 \times 10^{-2}$ represents the hundredths place, and $9 \times 10^{-3}$ represents the thousandths place.

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: 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: 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

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

[Solution Description]

The assertion states that the value of $5^{-2}$ is $\frac{1}{25}$. According to the rule of negative exponents, for any non-zero integer a, $a^{-m} = \frac{1}{a^m}$. Here, a is 5 and m is 2. So, $5^{-2} = \frac{1}{5^2} = \frac{1}{25}$. The reason is true and correctly explains the assertion.

Key Concept: Negative Exponents, General Form

c) $\frac{1}{32}$

[Solution Description]

Given the general form $a^{–m} = \frac{1}{a^m}$, substitute $a = 2$ and $m = 5$ to find $2^{–5}$. This gives $2^{–5} = \frac{1}{2^5}$. Calculating $2^5$ gives $32$, so $2^{–5} = \frac{1}{32}$.

Key Concept: Exponent Multiplication, Division

c) 4

[Solution Description]

To simplify $\frac{(4^2)^3}{4^5}$, we first simplify the numerator using the power of a power rule $(a^m)^n = a^{mn}$:

$(4^2)^3 = 4^{2 \times 3} = 4^6$

Now, the expression becomes:

$\frac{4^6}{4^5}$

Using the quotient rule $\frac{a^m}{a^n} = a^{m-n}$:

$\frac{4^6}{4^5} = 4^{6-5} = 4^1 = 4$

The simplified form of the expression is 4.

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

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

[Solution Description] By the Product Law of Exponents, for any non-zero integer $a$, the equation $a^m \times a^n = a^{m+n}$ is valid for all integers $m$ and $n$, whether positive or negative. This can be verified by taking examples such as $2^{-3} \times 2^{-2} = 2^{-5}$ or $(-3)^{-4} \times (-3)^{-3} = (-3)^{-7}$. Thus, both the Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Exponents, Product Law

b) 11

[Solution Description] Using the product law of exponents, we know that when multiplying two powers with the same base, their exponents are added. So, $x^{4} \times x^{7} = x^{4+7} = x^{11}$. Therefore, the value of $k$ is 11.

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: 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: Quotient Law of Exponents

b) $10^2$

[Solution Description]

To simplify $\frac{10^6}{10^4}$, we apply the quotient law of exponents, which states that $\frac{a^m}{a^n} = a^{m-n}$. Here, $a = 10$, $m = 6$, and $n = 4$. Substituting these values into the formula, we get $10^{6-4} = 10^2$. Therefore, the simplified value is $10^2$.

Key Concept: Quotient Law, Zero Exponent

c) $\frac{2}{3}$

[Solution Description]

To simplify $(2^0 \times 3^{-1}) \div 2^{-1}$, first evaluate $2^0$ which is equal to 1:

$2^0 = 1$

Now, multiply by $3^{-1}$, which is equal to $\frac{1}{3}$:

$1 \times \frac{1}{3} = \frac{1}{3}$

Finally, divide by $2^{-1}$, which is equal to 2:

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

Key Concept: Power of a Power Law

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

[Solution Description] The assertion states that $(3^2)^4 = 3^{8}$. Let's verify this using the Power of a Power Law. Applying the law, $(a^m)^n = a^{mn}$, we get $(3^2)^4 = 3^{2 \times 4} = 3^{8}$. Therefore, the assertion is true. The reason given is the statement of the Power of a Power Law, which is also true and correctly explains why $(3^2)^4 = 3^{8}$. Thus, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Power of a Power Law

b) $2^{-12}$

[Solution Description]

We are given the expression $(2^4)^{-3}$. According to the Power of a Power Law, $(a^m)^n = a^{mn}$. Applying this law:

$(2^4)^{-3} = 2^{4 \times (-3)} = 2^{-12}$

Therefore, the simplified expression is $2^{-12}$.

Key Concept: Power of a Power Law

b) $2^{12}$

[Solution Description]

To simplify $(2^4)^3$, we use the Power of a Power Law, which states that $(a^m)^n = a^{mn}$. Here, $a = 2$, $m = 4$, and $n = 3$. Applying the law gives $(2^4)^3 = 2^{4 \times 3} = 2^{12}$.

Key Concept: Product to Power Law

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

[Solution Description]

The assertion states that $(2 \times 5)^3$ simplifies to $8 \times 125$. Let's verify this using the product to power law. According to the law, $(ab)^m = a^m \times b^m$. Here, $a = 2$, $b = 5$, and $m = 3$. Applying the law, we get $(2 \times 5)^3 = 2^3 \times 5^3 = 8 \times 125$. Thus, the assertion is true. The reason correctly explains the assertion by stating the product to power law, which is directly applied to derive the result. Therefore, both the assertion and reason are true, and the reason is the correct explanation of the assertion.

Key Concept: Laws of Exponents: Product to Power Law

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

[Solution Description]

The Assertion claims that $a^{-m} \times a^{n} = a^{n - m}$ for any non-zero integer $a$ and integers $m$ and $n$. According to the Product to Power Law, $a^m \times a^n = a^{m + n}$, which is valid even when exponents are negative. Therefore, $a^{-m} \times a^{n} = a^{(-m) + n} = a^{n - m}$. Hence, both the Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Product to Power Law

a) $27$

[Solution Description]

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

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

Now, $3^3 = 27$, so the simplified form is $27$.

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

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, Fractional Exponents

b) $\frac{3}{4}$

[Solution Description]

To simplify $\left( \frac{9}{16} \right)^{\frac{1}{2}}$, we use the Quotient to Power Law. Apply the exponent to both numerator and denominator: $\left( \frac{9}{16} \right)^{\frac{1}{2}} = \frac{9^{\frac{1}{2}}}{16^{\frac{1}{2}}}$. Now, apply the exponent $\frac{1}{2}$ which is equivalent to taking the square root: $\frac{\sqrt{9}}{\sqrt{16}} = \frac{3}{4}$.

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, 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 is equal to 1. Here, the base is $1/2$, which is a non-zero number. Therefore, $(1/2)^0 = 1$.

Key Concept: Standard Form

c) $3.0 \times 10^{-6}$

[Solution Description]

To express 0.000003 m in standard form, we need to move the decimal point to the right until it is after the first non-zero digit. In this case, we move the decimal point 6 places to the right, which gives us 3.0. Since we moved the decimal point 6 places to the right, the exponent will be -6. Therefore, the standard form of 0.000003 m is $3.0 \times 10^{-6}$.

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 0.00000045 in standard form, we move the decimal point to the right until we get a number between 1 and 10. Here, we move the decimal point 7 places to the right, resulting in $4.5$. Since we moved the decimal point to the right, the exponent of 10 is negative, specifically $-7$. Therefore, the standard form of 0.00000045 is $4.5 \times 10^{-7}$. The reason correctly explains that for numbers less than 1, the standard form involves a decimal between 1 and 10 multiplied by 10 raised to a negative exponent.

Key Concept: Usual Form

c) 352000

[Solution Description]

To convert $3.52 \times 10^{5}$ into its usual form, we need to multiply 3.52 by 100000 (since $10^{5} = 100000$).

$3.52 \times 100000 = 352000$

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: Scientific Notation, Exponents

b) $5.45 \times 10^{-4}$

[Solution Description]

To find the ratio of the mass of an electron to the mass of a proton, divide the mass of the electron by the mass of the proton: $\frac{9.11 \times 10^{-31}}{1.67 \times 10^{-27}} = \frac{9.11}{1.67} \times 10^{-31 + 27} = 5.45 \times 10^{-4}$.

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 $0.000003$ in standard form, we move the decimal point to the right until it is after the first non-zero digit. This requires moving the decimal point 6 places to the right, giving us $3$. Since we moved the decimal point to the right, the exponent is negative. Therefore, $0.000003 = 3 \times 10^{-6}$. This confirms that Assertion (A) is true. Reason (R) states that in standard form, the exponent is negative when the original number is less than 1, which is correct because moving the decimal point to the right results in a negative exponent. Thus, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Negative exponents, Standard form

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

[Solution Description]

To convert the number $0.000007$ into standard form, we move the decimal point to the right until it is after the first non-zero digit. This requires moving the decimal point 6 places. Therefore, the exponent is $-6$. Thus, $0.000007 = 7 \times 10^{-6}$. This matches the assertion that $0.000007$ in standard form is $7 \times 10^{-6}$. The reason states that negative exponents are used to express very small numbers in standard form, which is a fundamental concept in mathematics. Hence, both the assertion and the reason are true, and the reason correctly explains the assertion.

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 $0.000007$ in standard form, we move the decimal point to the right until it is after the first non-zero digit, which is 7. We count the number of places the decimal has been moved, which is 6 places to the right. Therefore, the standard form is $7 \times 10^{-6}$. A negative exponent indeed indicates that the number is very small, and moving the decimal point to the right results in such an exponent. Thus, both the Assertion and Reason are true, and the Reason correctly explains the Assertion.

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: Standard Form, Distance from Earth to Sun

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. Converting this to standard form involves moving the decimal point to create a number between 1 and 10, resulting in $1.496 \times 10^{11}$ m. The reason correctly states that standard form is used to express very large numbers in a compact way. Both the assertion and the reason are true, and the reason correctly explains why the assertion is true.

Key Concept: Exponent Calculation

c) 11

[Solution Description]

The standard form representation of the distance from the Earth to the Sun is $1.496 \times 10^{11}$. Comparing this with $1.496 \times 10^{x}$, we find that x must be 11.

Key Concept: Assertion-Reason

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

[Solution Description]

To express $0.000007$ in standard form, we count the number of places the decimal point moves to the right until it is after the first non-zero digit. Here, the decimal point moves $6$ places to the right, so $0.000007 = 7 \times 10^{-6}$. The Reason explains that standard form uses powers of 10 to express very small numbers, which is correct. Therefore, both Assertion and Reason are true, and Reason correctly explains Assertion.

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 the diameter of a red blood cell is $7 \times 10^{-6}$ m. This is true because the average diameter of a red blood cell is 0.000007 m, which can be expressed in standard form as $7 \times 10^{-6}$ m. The reason states that the number $0.000007$ can be expressed as $7 \times 10^{-6}$. This is also true because $0.000007 = \frac{7}{1000000} = \frac{7}{10^6} = 7 \times 10^{-6}$. The reason correctly explains the assertion by showing how the small number is converted into its standard form.

Key Concept: Standard Form Conversion

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

[Solution Description]

To express 0.000007 m in standard form, we can write it as $0.000007 = \frac{7}{1000000} = \frac{7}{10^6} = 7 \times 10^{-6}$. Therefore, the standard form of 0.000007 m is $7 \times 10^{-6}$ m.

Key Concept: Comparing Large and Small Numbers

a) $5 \times 10^{-6}$ m

[Solution Description]

To find the difference in sizes between the red blood cell and the white blood cell, we subtract the size of the red blood cell from the size of the white blood cell.

$1.2 \times 10^{-5} - 7 \times 10^{-6} = 1.2 \times 10^{-5} - 0.7 \times 10^{-5} = (1.2 - 0.7) \times 10^{-5} = 0.5 \times 10^{-5} = 5 \times 10^{-6}$

So, the difference in their sizes is $5 \times 10^{-6}$ m.

Key Concept: Comparing Large and Small Numbers

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

[Solution Description]

To compare the diameters of the Sun and the Earth, we can use the given values:

Diameter of the Sun = $1.4 \times 10^9$ m

Diameter of the Earth = $1.2756 \times 10^7$ m

The ratio of the diameter of the Sun to the diameter of the Earth is:

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

Therefore, the diameter of the Sun is about 100 times the diameter of the Earth, meaning the diameter of the Earth is approximately 100 times smaller than the diameter of the Sun. The Assertion is true, and the Reason correctly explains the Assertion.

Key Concept: Comparing Large and Small Numbers

a) $2.568 \times 10^{-3}$

[Solution Description]

To find the ratio of the distance between Earth and Moon to the distance between Earth and Sun, we divide the distance between Earth and Moon by the distance between Earth and Sun.

$\frac{3.84 \times 10^{8}}{1.496 \times 10^{11}} = \frac{3.84}{1.496} \times 10^{8-11} = \frac{3.84}{1.496} \times 10^{-3}$

$\frac{3.84}{1.496} = 2.568$ (approx.). Therefore, $\frac{3.84}{1.496} \times 10^{-3} = 2.568 \times 10^{-3}$. So, the ratio is approximately $2.568 \times 10^{-3}$.

Key Concept: Exponent comparison

d) 1833

[Solution Description]

To find how many times heavier a proton is compared to an electron, divide the mass of the proton by the mass of the electron:

$\frac{1.67 \times 10^{-27}}{9.11 \times 10^{-31}} = \frac{1.67}{9.11} \times 10^{-27-(-31)} = \frac{1.67}{9.11} \times 10^{4} = 0.1833 \times 10^{4} = 1833$

Therefore, a proton is approximately 1833 times heavier than an electron.

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: 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 verify the assertion and reason, we can perform the division $\frac{1.4 \times 10^9}{1.2756 \times 10^7}$. Simplifying the exponents, we get $\frac{1.4 \times 10^{9-7}}{1.2756} = \frac{1.4 \times 10^2}{1.2756}$. Calculating this further, $\frac{140}{1.2756} \approx 100$. Thus, both the assertion and reason are true, and the reason correctly explains the 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: Addition and Subtraction in Standard Form

a) $2.6 \times 10^{-6}$

[Solution Description]

To subtract two numbers in standard form, ensure the exponents are the same. Here, both numbers have the same exponent of $10^{-6}$. Subtract the coefficients: 9.8 - 7.2 = 2.6. Thus, the result is $2.6 \times 10^{-6}$.

Key Concept: Standard Form, Addition

a) $3.84 \times 10^{-4}$

[Solution Description]

To add $3.6 \times 10^{-4}$ and $2.4 \times 10^{-5}$, first express both numbers with the same exponent. Convert $2.4 \times 10^{-5}$ to $0.24 \times 10^{-4}$. Now, add $3.6 \times 10^{-4}$ and $0.24 \times 10^{-4}$ to get $3.84 \times 10^{-4}$.

Key Concept: Converting exponents to the same power

b) $6.69 \times 10^{-3}$

[Solution Description]

First, convert $5.1 \times 10^{-4}$ to $0.51 \times 10^{-3}$. Now, subtract the numbers: $7.2 \times 10^{-3} - 0.51 \times 10^{-3} = 6.69 \times 10^{-3}$.

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: Exponents, Standard Form

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

[Solution Description]

Let us verify the assertion by converting $9^{-2}$ to a power with base 3. We know that $9 = 3^2$, so:

$9^{-2} = (3^2)^{-2} = 3^{2 \times (-2)} = 3^{-4}$

Thus, the assertion is true. The reason given is the law of exponents, which correctly explains how to convert the expression. Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

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: Standard Form Conversion

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

[Solution Description]

Standard form requires the coefficient to be between 1 and 10. Therefore, we adjust $604.35 \times 10^{22}$ to standard form:

$604.35 \times 10^{22} = 6.0435 \times 10^{2} \times 10^{22} = 6.0435 \times 10^{24} \text{ kg}$

Thus, the standard form is $6.0435 \times 10^{24}$ kg.

Key Concept: Comparing Masses

a) 81.22

[Solution Description]

To find how many times the mass of the Earth is greater than the mass of the Moon, divide the mass of the Earth by the mass of the Moon:

$\frac{5.97 \times 10^{24}}{7.35 \times 10^{22}} = \frac{5.97}{7.35} \times 10^{24-22} = \frac{5.97}{7.35} \times 10^{2} = 0.8122 \times 10^{2} = 81.22$

Therefore, the mass of the Earth is 81.22 times greater than the mass of the Moon.

Key Concept: Mass of Earth, Mass of Moon

c) 81.22

[Solution Description]

To find how many times the mass of Earth is greater than the mass of Moon, divide the mass of Earth by the mass of Moon:

$\frac{5.97 \times 10^{24}}{7.35 \times 10^{22}}$

Simplify the exponents:

$\frac{5.97}{7.35} \times 10^{24-22}$ = $\frac{5.97}{7.35} \times 10^2$

Calculate the division:

$\frac{5.97}{7.35} \approx 0.8122$

Multiply by $10^2$:

$0.8122 \times 10^2 = 81.22$

Key Concept: Standard Form, Multiplication

a) $1.2 \times 10^{-11}$

[Solution Description]

To find the product of $A$ and $B$, multiply the coefficients and add the exponents:

$(3 \times 10^{-5}) \times (4 \times 10^{-7}) = (3 \times 4) \times 10^{-5 + (-7)} = 12 \times 10^{-12}$.

Since the coefficient is greater than 10, convert it to standard form: $1.2 \times 10^{-11}$.

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: Multiplication in Standard Form

a) $1.6 \times 10^{-1}$

[Solution Description] To find the product of $3.2 \times 10^{-4}$ and $5 \times 10^{3}$, multiply the coefficients and add the exponents. First, multiply the coefficients: $3.2 \times 5 = 16$. Next, add the exponents: $-4 + 3 = -1$. Therefore, the product is $16 \times 10^{-1}$, which is equal to $1.6$.

Key Concept: Standard Form

a) $7 \times 10^{-6}$

[Solution Description] To convert $0.000007$ m to standard form, we move the decimal point to the right until it is after the first non-zero digit (7). We count the number of places we moved the decimal, which is 6. Since the original number is less than 1, the exponent will be negative. Thus, $0.000007 = 7 \times 10^{-6}$.

Key Concept: Exponents, Standard Form

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

[Solution Description]

Let us verify the assertion and reason step by step.

Step 1: The given distances are

Distance between Sun and Earth = $1.496 \times 10$${11}$$) m

Distance between Earth and Moon = $3.84 \times 10$${8}$$$ m.

Step 2: To find the distance between the Sun and the Moon during a solar eclipse, we subtract the distance between the Earth and the Moon from the distance between the Sun and the Earth. However, both distances must have the same exponent for subtraction.

Step 3: Convert $1.496 \times 10$${11}$$$ m to $1.496 \times 1000 \times 10$${8}$$$ m = $1496 \times 10$${8}$$$ m.

Step 4: Now subtract $3.84 \times 10$${8}$$$ m from $1496 \times 10$${8}$$$ m:

$1496 - 3.84) \times 10$${8}$$$ m = $1492.16 \times 10$${8}$$$ m.

Step 5: This can be written as $1.49216 \times 10$${11}$$$ m.

Thus, the assertion is true.

The reason explains the method used to calculate the distance between the Sun and the Moon by ensuring both distances have the same exponent before subtraction. Hence, the reason is true and correctly explains the assertion.

Key Concept: Distance between Sun and Moon, Scientific Notation

d) $10^{6}$ seconds

[Solution Description]

To find the time taken for the spacecraft to travel from the Sun to the Moon, use the formula: Time = Distance / Speed. Given: Distance = $1.49216 \times 10^{11}$m, Speed = $1.49216 \times 10^{5}$m/s. Calculate the time: $\frac{1.49216 \times 10^{11}}{1.49216 \times 10^{5}} = 10^{6}$ seconds.

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 between the Sun and the Moon is approximately $3.84 \times 10^8$ meters, which is correctly expressed in standard form. Standard form is indeed used to express very large or very small numbers conveniently. Therefore, both the Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Distance between Sun and Moon, Standard Form Conversion

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

[Solution Description]

To find the distance between the Sun and Moon, subtract the distance between Earth and Moon from the distance between the Sun and Earth. Given: Distance between Sun and Earth = $1.496 \times 10^{11}$m, Distance between Earth and Moon = $3.84 \times 10^{8}$m. First, express both distances with the same exponent: $1.496 \times 10^{11} = 1496 \times 10^{8}$. Now, subtract the distances: $(1496 - 3.84) \times 10^{8} = 1492.16 \times 10^{8}$m. Convert to standard form: $1.49216 \times 10^{11}$m.