Power of a Power Practice Problems with Solutions

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, we will use the Power of a Power rule of exponents, which simplifies expressions where an exponent is raised to another power. The rule is expressed as:

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

Now, let’s apply this rule to the given problem:

$(12^8)^4$

Step-by-step solution:

• Identify the base and exponents: In this case, the base is 12, with the first exponent being 8 and the second exponent being 4.
• Apply the Power of a Power rule by multiplying the exponents: $(8) \cdot (4) = 32$.
• Replace the original expression with the new exponent: $(12^8)^4 = 12^{32}$.

Therefore, the simplified expression is $\mathbf{12^{32}}$.

Let's compare the answer with the given choices:

• Choice 1: $12^4$ - Incorrect, uses incorrect exponent rule.
• Choice 2: $12^{12}$ - Incorrect, uses incorrect exponent multiplication.
• Choice 3: $12^2$ - Incorrect, unrelated solution.
• Choice 4: $12^{32}$ - Correct, matches our calculation.

Thus, the correct choice is Choice 4: $12^{32}$.

Therefore, the expression $(12^8)^4$ simplifies to $12^{32}$, confirming the correct choice is indeed Choice 4.

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, 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: $2^{2 \times 3}$.