Key Concept: Basic Concept

c) Speed = Distance / Time

[Solution Description]

Speed is calculated using the formula: $\text{Speed} = \frac{\text{Distance}}{\text{Time}}$. This formula shows that speed is a measure of how fast an object moves over a certain distance in a given time.

Key Concept: Introduction

c) Foundational concepts

[Solution Description] An introductory course primarily focuses on providing a foundation and basic understanding of a subject. It often covers key concepts, principles, and methodologies that are essential for further study. The goal is to familiarize students with the subject matter and prepare them for more advanced topics.

Key Concept: Basic Concept

c) Newton

[Solution Description]

The SI unit of force is the newton, which is represented by the symbol N. Force is defined as mass multiplied by acceleration, and the newton is derived from this relationship.

Key Concept: Definition of Exponents

d) 25

[Solution Description]

The expression $5^2$ represents 5 multiplied by itself 2 times. So, $5^2 = 5 \times 5 = 25$.

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

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

b) $10^4$

[Solution Description]

10,000 can be expressed as $10^4$ because $10 \times 10 \times 10 \times 10 = 10,000$.

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: 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, Large Numbers

d) 750,000,000,000,000,000,000,000 kg

[Solution Description]

The given mass is $7.5 \times 10^{23}$ kg. To write this in standard form, we move the decimal point 23 places to the right, adding zeros as necessary. This gives us 750,000,000,000,000,000,000,000 kg.

Key Concept: Understanding Powers, Exponents

d) 4

[Solution Description]

We are given the equation $3^{x} = 81$. To find the value of $x$, we need to express 81 as a power of 3. We know that $3^4 = 81$. Therefore, by comparing exponents, we get $x = 4$.

Key Concept: Exponents and Powers

b) 125

[Solution Description] To simplify $\frac{5^6}{5^3}$, we use the property of exponents that states $\frac{a^m}{a^n} = a^{m-n}$. Here, $a = 5$, $m = 6$, and $n = 3$. So, $\frac{5^6}{5^3} = 5^{6-3} = 5^3$. Now, calculate $5^3$ which is $5 \times 5 \times 5 = 125$.

Key Concept: Base and exponent notation

b) $\frac{5}{16}$

[Solution Description]

First, calculate each term separately:

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

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

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

Now, add and subtract these values:

$\frac{1}{8} + \frac{1}{4} - \frac{1}{16} = \frac{2}{16} + \frac{4}{16} - \frac{1}{16} = \frac{5}{16}$

Therefore, the value of $2^{-3} + 2^{-2} - 2^{-4}$ is $\frac{5}{16}$.

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

c) 250

[Solution Description]

To find $\frac{10^3}{2^2}$, first calculate the numerator and the denominator separately.

$10^3 = 10 \times 10 \times 10 = 1000$.

$2^2 = 2 \times 2 = 4$.

Divide the results: $\frac{1000}{4} = 250$.

Key Concept: Expanding expressions with exponents

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

[Solution Description]

To expand the number \$4506.39\$ using exponents, we represent each digit with its corresponding place value:

$4506.39 = 4 \times 10^3 + 5 \times 10^2 + 0 \times 10^1 + 6 \times 10^0 + 3 \times 10^{-1} + 9 \times 10^{-2}$

Here, \$10^{-1} = \frac{1}{10}\$ and \$10^{-2} = \frac{1}{100}\$. Hence, the expanded form is accurately captured by the equation above.

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) 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: 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: Exponents

a) 0.25

[Solution Description]

The negative exponent rule states that $a^{-n} = \frac{1}{a^{n}}$. Here, $a = 2$ and $n = 2$. So,

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

Therefore, the value of $2^{-2}$ is $\frac{1}{4}$.

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: Simplifying large and small numbers

a) $6 \times 10^8$

[Solution Description]

To find the product of $3 \times 10^5$ and $2 \times 10^3$, multiply the coefficients and add the exponents of 10.

The calculation is $(3 \times 2) \times 10^{5+3} = 6 \times 10^8$.

Therefore, the product is $6 \times 10^8$.

Key Concept: Identifying Very Small Numbers

c) 0.000003 m

[Solution Description]

A very small number is typically less than 1 and has several zeros after the decimal point. Among the given options, 0.000003 m is the smallest number.

Key Concept: Converting to usual form

c) 5,800,000,000,000

[Solution Description]

To convert $5.8 \times 10^{12}$ to its usual form, we need to move the decimal point 12 places to the right since the exponent is positive. Start with 5.8 and add zeros as needed: 5.8 becomes 5800000000000. Therefore, the usual form is 5,800,000,000,000.

Key Concept: Negative Exponents

b) 9

[Solution Description]

To solve $(\frac{1}{3})^{-2}$, we can use the property of negative exponents which states that $a^{-n} = \frac{1}{a^n}$. Applying this, we get:

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

Therefore, the value is 9.

Key Concept: Negative Exponents, Evaluation

b) 5

[Solution Description]

First, compute $3^0$ which is equal to 1. Next, compute $4^{-1}$ using the rule $a^{-n} = \frac{1}{a^n}$, which gives:

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

Now add these two results:

$3^0 + 4^{-1} = 1 + \frac{1}{4} = \frac{5}{4}$

Multiply this sum by $2^2$, which is 4:

$\left(\frac{5}{4}\right) \times 4 = 5$

So the final answer is 5.

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: Negative Exponents, Real-World Application

b) 1 gram

[Solution Description]

The formula for the amount of substance remaining after time $t$ is given by $A = 1000 \times 10^{-t}$. We need to find the amount remaining after 3 hours, so substitute $t = 3$ into the formula: $A = 1000 \times 10^{-3}$. Calculate $10^{-3} = \frac{1}{1000} = 0.001$. Therefore, $A = 1000 \times 0.001 = 1$ gram.

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: 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: Expanding Numbers Using Exponents

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

[Solution Description]

The expanded form of 1025.63 using exponents is $1 \times 10^3 + 0 \times 10^2 + 2 \times 10^1 + 5 \times 10^0 + 6 \times 10^{-1} + 3 \times 10^{-2}$.

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: Powers with Negative Exponents, Multiplicative Inverse Using Exponents

a) 3

[Solution Description]

Given $x^{–4} = \frac{1}{81}$, we can rewrite it using the property $a^{–m} = \frac{1}{a^m}$:

$x^{–4} = \frac{1}{x^4} = \frac{1}{81}$

This implies:

$x^4 = 81$

To find $x$, take the fourth root of both sides:

$x = \sqrt[4]{81}$

Since $3^4 = 81$, we get:

$x = 3$

Key Concept: Expanding Numbers Using Negative Exponents

c) 0.5

[Solution Description]

The expression $5 \times 10^{-1}$ can be simplified by understanding that $10^{-1}$ is equal to $\frac{1}{10}$. Therefore, $5 \times 10^{-1} = 5 \times \frac{1}{10} = 0.5$.

Key Concept: Negative Exponents, Expanded Form

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

[Solution Description]

To expand 1025.63 using exponents, we break it down into its place values:

$1025.63 = 1 \times 10^3 + 0 \times 10^2 + 2 \times 10^1 + 5 \times 10^0 + 6 \times 10^{-1} + 3 \times 10^{-2}$

Thus, the expanded form is $1 \times 10^3 + 0 \times 10^2 + 2 \times 10^1 + 5 \times 10^0 + 6 \times 10^{-1} + 3 \times 10^{-2}$.

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) 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

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: 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

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: 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: Laws of Exponents

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

[Solution Description] The assertion states the law of exponents for multiplying powers with the same base, which is $a^m \times a^n = a^{m + n}$. The reason correctly explains that this law holds true because multiplying two powers with the same base results in the exponents being added together. Both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Product Law

b) $y^{11}$

[Solution Description]

Using the product law of exponents, when multiplying terms with the same base, add the exponents. So, $y^4 \cdot y^7 = y^{4+7} = y^{11}$.

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: 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: Quotient Law of Exponents

b) $3^3$

[Solution Description]

To simplify $\frac{3^8}{3^5}$, we apply the quotient law of exponents, which states that $\frac{a^m}{a^n} = a^{m-n}$. Here, $a = 3$, $m = 8$, and $n = 5$. Substituting these values into the formula, we get $3^{8-5} = 3^3$. Therefore, the simplified value is $3^3$.

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

c) 25

[Solution Description]

Using the Quotient Law of Exponents, $\frac{a^m}{a^n} = a^{m-n}$, with $a = 5$, $m = 8$, and $n = 6$. Thus:

$\frac{5^8}{5^6} = 5^{8-6} = 5^2$

Calculating $5^2$:

$5^2 = 25$

The simplified form is $25$.

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: Power of a Power Law

b) $3^8$

[Solution Description]

First, apply the power of a power law, $(a^m)^n = a^{mn}$, to simplify $((3^2)^{-4})^{-1}$. Start with the innermost exponent: $(3^2)^{-4} = 3^{2 \times (-4)} = 3^{-8}$. Now, apply the power of a power law again to simplify the outer exponent: $(3^{-8})^{-1} = 3^{(-8) \times (-1)} = 3^8$. Therefore, the simplified form is $3^8$.

Key Concept: Power of a Power Law

b) $3^8$

[Solution Description]

To simplify $(3^2)^4$, we use the Power of a Power Law, which states that $(a^m)^n = a^{mn}$. Here, $a = 3$, $m = 2$, and $n = 4$. Applying the law gives $(3^2)^4 = 3^{2 \times 4} = 3^8$.

Key Concept: Product to Power Law

c) 15625

[Solution Description]

Using the Product to Power Law, $(a^m)^n = a^{mn}$. Therefore:

$(5^2)^3 = 5^{2 \times 3} = 5^6 = 15625$

Key Concept: Product to Power Law

b) 6561

[Solution Description]

Applying the Product to Power Law, $(a^m)^n = a^{mn}$, we have:

$(3^4)^2 = 3^{4 \times 2} = 3^8 = 6561$

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: Quotient to Power Law

a) $\frac{p^{12}}{q^6}$

[Solution Description]

To simplify $(\frac{p^4}{q^2})^3$, apply the Quotient to Power Law, which states $(\frac{x}{y})^n = \frac{x^n}{y^n}$:

$(\frac{p^4}{q^2})^3 = \frac{(p^4)^3}{(q^2)^3} \\ ] \\ Next, use the Power of a Power Law, $(x^m)^n = x^{m \cdot n}$, to simplify the exponents:

$\frac{(p^4)^3}{(q^2)^3} = \frac{p^{4 \cdot 3}}{q^{2 \cdot 3}} = \frac{p^{12}}{q^6}$

The simplified form is $\frac{p^{12}}{q^6}$.

Key Concept: Quotient to Power Law

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

[Solution Description]

To verify the assertion, we apply the Quotient to Power Law:

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

The assertion is true as it correctly follows the law. The reason is also true because it states the correct mathematical rule used in the simplification. Therefore, both the assertion and reason are true, and the reason explains the assertion correctly.

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, Fractional Bases

c) $1 + \sqrt{z} + \frac{1}{\sqrt{z}}$

[Solution Description]

Given that $z$ is a non-zero integer, using the Zero Exponent Rule, $z^0 = 1$. Also, $z^{\frac{1}{2}} = \sqrt{z}$ and $z^{-\frac{1}{2}} = \frac{1}{\sqrt{z}}$. Therefore, $z^0 + z^{\frac{1}{2}} + z^{-\frac{1}{2}} = 1 + \sqrt{z} + \frac{1}{\sqrt{z}}$. Since no specific value of $z$ is given, the expression remains in this form.

Key Concept: Zero Exponent Rule

b) 1

[Solution Description]

The expression $(a^3)^0$ can be simplified using the Zero Exponent Rule. Regardless of the value of $a^3$, as long as it is not zero, raising it to the power of 0 will result in 1. Given that $a = 5$, which is a non-zero number, $(5^3)^0 = 1$.

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 $1/2$, which is a non-zero number. Therefore, $(1/2)^0 = 1$.

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: Standard Form

c) $5.0 \times 10^{-3}$

[Solution Description]

To express 0.005 cm 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 3 places to the right, which gives us 5.0. Since we moved the decimal point 3 places to the right, the exponent will be -3. Therefore, the standard form of 0.005 cm is $5.0 \times 10^{-3}$.

Key Concept: Assertion-Reason

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

[Solution Description] To express the number $0.000007$ in standard form, we need to write it as a number between 1 and 10 multiplied by a power of 10. Moving the decimal point to the right until it is after the first non-zero digit gives $7$. Since we moved the decimal point 6 places to the right, we multiply by $10^{-6}$. Thus, $0.000007 = 7 \times 10^{-6}$. The reason correctly explains the process of converting very small numbers to standard form.

Key Concept: Scientific Notation

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

[Solution Description]

The assertion states that $0.000003$ can be expressed in standard form as $3 \times 10^{-6}$. Let's verify this step by step. First, identify the number $0.000003$. To convert it into standard form, we need to move the decimal point to the right until there is only one non-zero digit before the decimal point. In this case, we move the decimal point 6 places to the right to get $3.0$. Since we moved the decimal point 6 places to the right, the exponent of 10 will be $-6$. Therefore, the standard form is $3 \times 10^{-6}$. The reason explains that to express a small number in standard form, we move the decimal point to the right until there is only one non-zero digit before the decimal point and count the number of places moved, which is then used as the negative exponent of 10. This is also correct and directly explains the assertion. Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Scientific Notation

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 first identify the decimal point and count the number of places it needs to move to the right to reach the first non-zero digit, which is 7 in this case. This gives us an exponent of $-6$. Therefore, $0.000007 = 7 \times 10^{-6}$. This matches the format of standard form for very small numbers, which is $a \times 10^{-n}$, where $1 \leq a < 10$ and $n$ is a positive integer. Both the Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Scientific Notation

b) $1.6 \times 10^{-3}$ cm

[Solution Description]

To express $0.0016$ cm in standard form, we move the decimal point to the right until it is after the first non-zero digit (1). This requires moving the decimal point 3 places to the right. Therefore, $0.0016 = 1.6 \times 10^{-3}$ cm.

Key Concept: Standard Form, Negative Exponents

d) $7 \times 10^{-3}$ mm

[Solution Description]

To express 0.007 mm in standard form, we need to move the decimal point to the right until it is after the first non-zero digit. Here, we move the decimal point 3 places to the right: $0.007 = 7 \times 10^{-3}$. Therefore, the standard form is $7 \times 10^{-3}$ mm.

Key Concept: Exponents, Standard Form

c) $0$ coulomb

[Solution Description]

The charge of an electron is $-1.6 \times 10^{-19}$ coulomb, and the charge of a proton is $1.6 \times 10^{-19}$ coulomb. Therefore, the total charge for $10^{20}$ electrons is:

$$

10^{20} \times (-1.6 \times 10^{-19}) = -16 \text{ coulomb}

$$

The total charge for $10^{20}$ protons is:

$$

10^{20} \times (1.6 \times 10^{-19}) = 16 \text{ coulomb}

$$

Adding both charges together:

$$

-16 \text{ coulomb} + 16 \text{ coulomb} = 0 \text{ coulomb}

$$

Therefore, the total charge is $0$ coulomb.

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: Comparing Magnitudes

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

[Solution Description]

The distance from the Earth to the Sun in standard form is $1.496 \times 10^{11}$. This is the same as 149,600,000,000 meters.

Key Concept: Exponent Calculation

c) 11

[Solution Description]

Given the standard form of the distance from Earth to Sun: $1.496 \times 10^{11}$ m.

1. Compare with the given expression $1.496 \times 10^{n}$ m.

2. Equate the exponents: $n = 11$.

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]

The assertion states that the diameter of a Red Blood Cell is $7 \times 10^{-6}$ m. According to the syllabus, 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. Therefore, the assertion is true. The reason states that the standard form of $0.000007$ m is $7 \times 10^{-6}$ m, which is also true according to the syllabus. Furthermore, the reason correctly explains why the assertion is true. Hence, both the assertion and reason are true, and the reason is the correct explanation of the assertion.

Key Concept: Comparison of Sizes

a) $\frac{1}{2}$

[Solution Description]

The ratio of the size of the red blood cell to the plant cell is given by:

$\frac{\text{Size of Red Blood Cell}}{\text{Size of Plant Cell}} = \frac{7 \times 10^{-6}}{1.275 \times 10^{-5}}$

Simplify the expression by dividing the coefficients and subtracting the exponents:

$\frac{7}{1.275} \times 10^{-6 - (-5)} = \frac{7}{1.275} \times 10^{-1}$

$\frac{7}{1.275} = 5.49$, so the ratio is approximately $5.49 \times 10^{-1} = 0.55$

Therefore, the ratio of the size of the red blood cell to the plant cell is approximately $\frac{1}{2}$.

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: 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: 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: Use of Exponents to Express Small Numbers in Standard Form, Comparing Large and Small Numbers

d) Assertion is false, but Reason is true.

[Solution Description]

To verify the assertion, we need to compare the diameters of the Sun and the Earth using their given values in standard form. Let’s calculate the ratio of the diameters:

$\frac{\text{Diameter of the Sun}}{\text{Diameter of the Earth}} = \frac{1.4 \times 10^9}{1.2756 \times 10^7} = \frac{1.4}{1.2756} \times 10^{9-7} = \frac{1.4}{1.2756} \times 100 \approx 100$

The ratio is approximately 100, meaning the diameter of the Sun is about 100 times the diameter of the Earth. This contradicts the assertion, which states that the diameter of the Sun is approximately 10 times the diameter of the Earth. However, the reason correctly provides the diameters of the Sun and Earth, which are true values.

Therefore, the assertion is false, but the reason is true.

Key Concept: Comparing very large and very small numbers

d) $1.2756 \times 10^9$ m

[Solution Description]

Given that the diameter of the Sun is approximately 100 times the diameter of the Earth, we can calculate the diameter of the Sun by multiplying the diameter of the Earth by 100.

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

Diameter of the Sun = $1.2756 \times 10^7 \times 100$

Multiplying $1.2756 \times 10^7$ by 100 is equivalent to adding 2 to the exponent:

$1.2756 \times 10^{7+2} = 1.2756 \times 10^9$

Therefore, the diameter of the Sun is $1.2756 \times 10^9$ meters.

Key Concept: Comparing very large and very small numbers

b) 100

[Solution Description]

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

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

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

The ratio is calculated as:

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

Simplifying further:

$\frac{1.4 \times 100}{1.2756} \approx 100$

So, the diameter of the Sun is about 100 times the diameter of the Earth.

Key Concept: Use of Exponents, Standard Form

a) $4.634 \times 10^{6}$

[Solution Description]

To find the difference in the radii of the Earth and the Moon, we subtract their values:

$6.371 \times 10^{6} - 1.737 \times 10^{6} = (6.371 - 1.737) \times 10^{6} = 4.634 \times 10^{6}$

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) $7.7 \times 10^{-7}$

[Solution Description]

To add two numbers in standard form, ensure the exponents are the same. Here, both numbers have the same exponent of $10^{-7}$. Add the coefficients: 4.5 + 3.2 = 7.7. Thus, the result is $7.7 \times 10^{-7}$.

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

[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: Converting exponents to the same power before performing operations

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

[Solution Description]

The Assertion is true because when adding or subtracting numbers in scientific notation, they must have the same exponent to ensure that the digits are aligned correctly. The Reason is also true because converting exponents to the same power aligns the digits, making addition or subtraction straightforward. Moreover, the Reason correctly explains why the Assertion is true.

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, 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: 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: Exponents and Standard Form

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

[Solution Description]

Step 1: Mass of Earth is given as $5.97 \times 10^{24}$ kg. Mass of Moon is given as $7.35 \times 10^{22}$ kg.

Step 2: Convert the mass of Earth to the same power of 10 as the mass of Moon for addition.

Step 3: $5.97 \times 10^{24}$ kg can be written as $597 \times 10^{22}$ kg.

Step 4: Now, add the converted mass of Earth and the mass of Moon: $(597 + 7.35) \times 10^{22}$.

Step 5: Calculate the sum: $597 + 7.35 = 604.35$.

Step 6: Thus, the total mass is $604.35 \times 10^{22}$ 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: 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 and Division in Standard Form

b) $12 \times 10^{0}$

[Solution Description] First, multiply the numerators: $6 \times 10^{-2} \times 4 \times 10^{3} = 24 \times 10^{1}$. Then, divide by the denominator: $\frac{24 \times 10^{1}}{2 \times 10^{1}} = 12 \times 10^{0} = 12$.

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

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

[Solution Description] To express $0.00001275$ m in standard form, move the decimal point to the right until it is after the first non-zero digit (1). We moved the decimal point 5 places. Since the original number is less than 1, the exponent will be negative. Thus, $0.00001275 = 1.275 \times 10^{-5}$.

Key Concept: Standard Form Conversion

c) $0.384 \times 10^{9}$ m

[Solution Description]

To convert $3.84 \times 10^{8}$ m into standard form with an exponent of 9, we move the decimal one place to the left and adjust the exponent accordingly. Thus, $3.84 \times 10^{8}$ m = $0.384 \times 10^{9}$ m.

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 assertion states that the distance between the Sun and the Moon is approximately $3.84 \times 10^8$ meters. This is a factual statement based on astronomical data. The reason explains that this distance is expressed in standard form to simplify calculations and comparisons. Standard form is indeed used to represent very large or very small numbers in a concise and manageable way, making it easier to perform mathematical operations and comparisons. Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

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: $\frac{1.49216}{3.84} \times 10^{11-8} = 0.38857 \times 10^{3} = 388.57$.