Power of a Power Practice Problems with Solutions

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, we will proceed with the following steps:

• Identify the expression structure.
• Apply the power of a power rule for exponents.
• Simplify the expression.

Now, let's work through each step in detail:

Step 1: Identify the expression structure.
We have the expression $(10^3)^3$. This indicates a power of a power where the base is 10, the inner exponent is 3, and the entire expression is raised to another power of 3.

Step 2: Apply the power of a power rule.
The rule states $(a^m)^n = a^{m \times n}$. Applying this to our specific expression gives us:

Step 3: Perform the multiplication in the exponent.
Calculating $3 \times 3$, we get $9$. Thus, the expression simplifies to:

Therefore, the solution to the problem is:

Examining the provided choices:

• Choice 1: $10^{3+3}$ - Incorrect, because it uses addition instead of multiplication.
• Choice 2: $10^{3 \times 3}$ - Correct, as it matches our derived expression.
• Choice 3: $10^{\frac{3}{3}}$ - Incorrect, because it uses division instead of multiplication.
• Choice 4: $10^{3-3}$ - Incorrect, because it uses subtraction instead of multiplication.

The correct answer is $10^{3 \times 3}$, which is represented by Choice 2.

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.

We are given the expression $(2^2)^3$ and need to simplify it using the laws of exponents and identify the corresponding expression among the choices.

To simplify the expression $(2^2)^3$, we use the "power of a power" rule, which states that $(a^m)^n = a^{m \times n}$.

Applying this rule to our expression, we have:

Calculating the new exponent:

Thus, the expression simplifies to:

Now, let's compare our result $2^6$ with the given choices:

• Choice 1: $2^{2+3} = 2^5$ - Incorrect, as our expression evaluates to $2^6$, not $2^5$.
• Choice 2: $2^{2-3} = 2^{-1}$ - Incorrect, as our expression evaluates to $2^6$, not $2^{-1}$.
• Choice 3: $2^{\frac{2}{3}}$ - Incorrect, as our expression evaluates to $2^6$, not a fractional exponent expression.
• Choice 4: $2^{2 \times 3} = 2^6$ - Correct, as this matches our simplified expression.

Therefore, the correct choice is Choice 4: $2^{2 \times 3}$.