Power of a Power Practice Problems with Solutions

To simplify the expression $\left(8^5\right)^{10}$, we'll apply the power of a power rule for exponents.

• Step 1: Identify the given expression.
• Step 2: Apply the power of a power rule, which states that $(a^m)^n = a^{m \cdot n}$.
• Step 3: Multiply the exponents to simplify the expression.

Now, let's work through each step:
Step 1: The expression given is $\left(8^5\right)^{10}$.
Step 2: We will use the power of a power rule: $(a^m)^n = a^{m \cdot n}$.
Step 3: Multiply the exponents: $5 \cdot 10 = 50$.

Thus, the expression simplifies to $8^{50}$.

The correct simplified form of the expression $\left(8^5\right)^{10}$ is $8^{50}$, which corresponds to choice 2.

Alternative choices:

• Choice 1: $8^{15}$ is incorrect because it misapplies the exponent multiplication.
• Choice 3: $8^5$ is incorrect because it does not apply the power of a power rule.
• Choice 4: $8^2$ is incorrect and unrelated to the operation.

I am confident in the correctness of this solution.

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:

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

To solve the expression $(16^6)^7$, we will use the power of a power rule for exponents. This rule states that when you raise a power to another power, you multiply the exponents. Here are the steps:

• Identify the components: The base is 16, and the inner exponent is 6. The outer exponent is 7.
• Apply the power of a power rule: According to the rule, $(a^m)^n = a^{m \cdot n}$. Thus, $(16^6)^7 = 16^{6 \cdot 7}$.
• Multiply the exponents: Calculate the product of the exponents $6 \times 7$. This gives us 42.
• Rewrite the expression: Substitute the product back into the expression, giving us $16^{42}$.

Therefore, the simplified expression is $\mathbf{16^{42}}$.

Checking against the answer choices, we find:
1. $16^{42}$ is given as choice 1.
2. Other choices do not match the simplified expression.
Choice 1 is correct because it accurately reflects the application of exponent rules.

Consequently, we conclude that the correct solution is $\mathbf{16^{42}}$.

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 follow these steps:

• Step 1: Identify the given information.
• Step 2: Apply the appropriate exponent rule.
• Step 3: Perform the necessary calculations.

Let's work through each step:

Step 1: The given expression is $\left(2^7\right)^5$. Here, the base is $2$, and we have two exponents: $7$ in the inner expression and $5$ outside.

Step 2: We'll use the power of a power rule for exponents, which states $(a^m)^n = a^{m \cdot n}$. This means we will multiply the exponents $7$ and $5$.

Step 3: Calculating, we multiply the exponents:
$7 \times 5 = 35$

Therefore, the expression $\left(2^7\right)^5$ simplifies to $2^{35}$.

Now, let's verify with the given answer choices:

• Choice 1: $2^{12}$ - Incorrect, as the exponents were not multiplied properly.
• Choice 2: $2^2$ - Incorrect, as it significantly underestimates the combined exponent value.
• Choice 3: $2^{35}$ - Correct, matches the calculated exponent.
• Choice 4: $2^{\frac{5}{7}}$ - Incorrect, involves incorrect fraction of exponents.

Thus, the correct choice is Choice 3: $2^{35}$.

I am confident in the correctness of this solution as it directly applies well-established exponent rules.