Power of a Power Practice Problems with Solutions

To solve the exercise we use the power property:$(a^n)^m=a^{n\cdot m}$

We use the property with our exercise and solve:

$(3^5)^4=3^{5\times4}=3^{20}$

We use the formula:

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

Therefore, we obtain:

$6^{2\times13}=6^{26}$

To solve this problem, we need to simplify the expression $\left(6^2\right)^7$ using the power of a power rule.

The power of a power rule states that when you have an expression of the form $(a^m)^n$, this can be simplified to $a^{m \times n}$.

Let's apply this rule to the given expression:

1. Identify the base and exponents: - Base: $6$ - First exponent (inside parenthesis): $2$ - Second exponent (outside parenthesis): $7$

2. Apply the power of a power rule: - Simplify $(6^2)^7 = 6^{2 \times 7}$.

3. Calculate the final exponent: - Multiply the exponents: $2 \times 7 = 14$. - Therefore, the simplified expression is $6^{14}$.

Considering the answer choices provided:

• Choice 1: $6^{2 \times 7}$ (Correct, as per our solution).
• Choice 2: $6^{2 + 7}$ (Incorrect, addition is used instead of multiplication).
• Choice 3: $6^{7-2}$ (Incorrect, subtraction is used incorrectly).
• Choice 4: $6^{\frac{7}{2}}$ (Incorrect, division is used incorrectly).

Thus, the correct answer to the problem is $6^{2 \times 7}$, which simplifies to $6^{14}$, and aligns with Choice 1.

To solve this problem, let's carefully follow these steps:

• Step 1: Identify the base and exponents in the expression.
• Step 2: Use the power of a power rule to simplify the expression.
• Step 3: Choose the appropriate option from the given answer choices.

Now, let's break this down:

Step 1: The expression given is $(4^5)^2$. Here, the base is 4, the inner exponent is 5, and the outer exponent is 2.

Step 2: We apply the power of a power rule for exponents, which states that $(a^m)^n = a^{m \cdot n}$.

Using the rule, we have:

This means the expression $(4^5)^2$ can be simplified to $4^{10}$.

Step 3: From the answer choices provided, we need to select the one corresponding to $4^{5 \cdot 2}$:

• Choice 1: $4^{\frac{2}{5}}$ - This is incorrect because it deals with division of exponents and not multiplication.
• Choice 2: $4^{5-2}$ - This is incorrect as it incorrectly subtracts the exponents.
• Choice 3: $4^{5 \times 2}$ - This is the correct choice.
• Choice 4: $4^{5+2}$ - This is incorrect as it incorrectly adds the exponents.

Therefore, the solution to the problem is $4^{5 \times 2} = 4^{10}$, which corresponds to choice 3.

To solve this problem, we'll utilize the Power of a Power rule of exponents, which states:

$(a^m)^n = a^{m \cdot n}$

Given the expression $(3^2)^4$, we need to simplify this by applying the rule:

• Step 1: Recognize that we have a base of 3, with an exponent of 2, raised to another exponent of 4.
• Step 2: According to the Power of a Power rule, we multiply the exponents: $2 \times 4$.
• Step 3: Compute the product of the exponents: $2 \times 4 = 8$.
• Step 4: Rewrite the expression as a single power: $3^8$.

This simplifies the original expression $(3^2)^4$ to $3^{8}$.

Comparing this with the given choices:

• Choice 1: $3^{2 \times 4}$ is equivalent to $3^8$, confirming it matches our solution.
• Choices 2, 3, and 4 involve incorrect operations with exponents (addition, subtraction, division) and therefore do not align with the necessary Power of a Power rule.

Thus, the correct answer to the problem is:

$3^{8}$, and this corresponds to Choice 1: $3^{2 \times 4}$.