Exponents - Power of a Power Rule worksheet with practice problems for simplifying expressions and solving for variables.

Problem Analysis:

The task involves solving problems related to exponents and the power of a power rule. Let's break it down step by step.

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#### Part 1: Simplify Each Expression Using the Power of a Power Rule

The Power of a Power Rule states:
\[
(a^m)^n = a^{m \cdot n}
\]

We will apply this rule to each expression.

1. Expression: \( (4^3)^2 \)
\[
(4^3)^2 = 4^{3 \cdot 2} = 4^6
\]

2. Expression: \( (a^5)^3 \)
\[
(a^5)^3 = a^{5 \cdot 3} = a^{15}
\]

3. Expression: \( (9^0)^7 \)
Recall that any number raised to the power of 0 is 1:
\[
9^0 = 1
\]
Therefore:
\[
(9^0)^7 = 1^7 = 1
\]

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#### Part 2: Find the Value of \( x \)

We are given equations involving exponents, and we need to solve for \( x \).

1. Equation: \( 3^x = 3^5 \)
Since the bases are the same, the exponents must be equal:
\[
x = 5
\]

2. Equation: \( 5^{x+1} = 5^3 \)
Again, since the bases are the same, the exponents must be equal:
\[
x + 1 = 3
\]
Solving for \( x \):
\[
x = 3 - 1 = 2
\]

3. Equation: \( 2^{x-2} = 2^4 \)
Similarly, since the bases are the same, the exponents must be equal:
\[
x - 2 = 4
\]
Solving for \( x \):
\[
x = 4 + 2 = 6
\]

4. Equation: \( 7^{2x} = 7^8 \)
Since the bases are the same, the exponents must be equal:
\[
2x = 8
\]
Solving for \( x \):
\[
x = \frac{8}{2} = 4
\]

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#### Part 3: Which of the Following Equals \( (2^3)^2 \)?

First, simplify \( (2^3)^2 \) using the Power of a Power Rule:
\[
(2^3)^2 = 2^{3 \cdot 2} = 2^6
\]

Now, evaluate each option:

- Option A: \( 2^5 \times 2 \)
\[
2^5 \times 2 = 2^5 \times 2^1 = 2^{5+1} = 2^6
\]
This matches \( 2^6 \).

- Option B: \( 2^3 \times 2^2 \)
\[
2^3 \times 2^2 = 2^{3+2} = 2^5
\]
This does not match \( 2^6 \).

- Option C: \( 2^4 \times 2^2 \)
\[
2^4 \times 2^2 = 2^{4+2} = 2^6
\]
This matches \( 2^6 \).

- Option D: \( 2^7 \div 2 \)
\[
2^7 \div 2 = 2^{7-1} = 2^6
\]
This matches \( 2^6 \).

Thus, the correct answers are A, C, and D.

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#### Part 4: Find the Value of \( 4^3 \times 4^{-3} \)

Use the property of exponents:
\[
a^m \times a^n = a^{m+n}
\]
Here:
\[
4^3 \times 4^{-3} = 4^{3 + (-3)} = 4^0
\]
Recall that any non-zero number raised to the power of 0 is 1:
\[
4^0 = 1
\]

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Final Answers:

1. Simplified expressions:
\[
(4^3)^2 = 4^6, \quad (a^5)^3 = a^{15}, \quad (9^0)^7 = 1
\]

2. Values of \( x \):
\[
x = 5, \quad x = 2, \quad x = 6, \quad x = 4
\]

3. Equivalents of \( (2^3)^2 \):
\[
\boxed{\text{A, C, D}}
\]

4. Value of \( 4^3 \times 4^{-3} \):
\[
\boxed{1}
\]