Power of a Power - Examples, Exercises and 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, 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: $3^{2 \times 4}$.

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

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 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 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.

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 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 this problem, we will apply the exponent rule for power of a power.

Let's go through the solution step-by-step:

• Step 1: Identify the given expression, which is $(7^8)^9$.
• Step 2: Apply the power of a power rule. This rule states that $(a^m)^n = a^{m \times n}$. In this case, $a = 7$, $m = 8$, and $n = 9$.
• Step 3: Calculate $8 \times 9$, which equals 72.
• Step 4: Rewrite the expression using the result: $(7^8)^9 = 7^{72}$.

Therefore, the simplified expression is $7^{72}$.

Looking at the answer choices, the correct choice is:

• Choice 1: $7^{72}$

This choice corresponds exactly with our solution. The other choices do not represent the simplified form of the original expression, making them incorrect.

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 the problem, we will simplify the expression $\left(\left(2 \times 3\right)^2\right)^5$ using the power of a power exponent rule. Follow these steps:

• Step 1: Identify the form of the expression. The given expression is $\left(\left(2 \times 3\right)^2\right)^5$.
• Step 2: Apply the power of a power rule, which states that $(a^m)^n = a^{m \times n}$.
• Step 3: Here, the base is $2 \times 3$, the first exponent ($m$) is 2, and the second exponent ($n$) is 5.
• Step 4: Multiply the exponents: $2 \times 5 = 10$.

Therefore, the expression simplifies to $\left(2 \times 3\right)^{10}$. However, for the purpose of matching the form requested, it can be expressed as $\left(2 \times 3\right)^{2 \times 5}$.

Next, we evaluate the given choices:

• Choice 1: $\left(2 \times 3\right)^{2+5}$ — This incorrectly adds the exponents instead of multiplying them.
• Choice 2: $\left(2 \times 3\right)^{5-2}$ — This incorrectly subtracts the exponents.
• Choice 3: $\left(2 \times 3\right)^{2\times5}$ — This correctly multiplies the exponents, which we found is the right simplification.
• Choice 4: $\left(2 \times 3\right)^{\frac{5}{2}}$ — This introduces division of exponents, which is not applicable here.

The correct choice is Choice 3: $\left(2 \times 3\right)^{2\times5}$.

To solve this problem, we'll use the power of a power property of exponents, which states that for any base $a$ and exponents $m$ and $n$, $(a^m)^n = a^{m \times n}$.

• Step 1: Identify the base and exponents:
  In the given expression $\left(\left(4 \times 6\right)^3\right)^4$, the base is $(4 \times 6)$, the inner exponent is 3, and the outer exponent is 4.

• Step 2: Apply the power of a power rule:
  According to the rule, $\left((4 \times 6)^3\right)^4$ simplifies to $(4 \times 6)^{3 \times 4}$.

• Step 3: Calculate the new exponent:
  Multiply the exponents: $3 \times 4 = 12$. Hence, the expression simplifies to $(4 \times 6)^{12}$.

The expression $\left(\left(4 \times 6\right)^3\right)^4$ is equivalent to $(4 \times 6)^{3 \times 4}$. Therefore, the correct choice is:

$\left(4\times6\right)^{3\times4}$

Therefore, the correct answer is Choice 1.

To solve the problem, we need to simplify the expression $\left(\left(3\times8\right)^5\right)^6$.

We will utilize the "power of a power" rule in exponents, which states $(a^m)^n = a^{m \times n}$. This rule tells us to multiply the exponents when raising a power to another power.

• Step 1: Identify the expression to simplify: $\left(\left(3 \times 8\right)^5\right)^6$.
• Step 2: Apply the power of a power rule: This gives us $(3 \times 8)^{5 \times 6}$.
• Step 3: Multiply the exponents: $5 \times 6 = 30$.

Therefore, the expression simplifies to $(3 \times 8)^{30}$.

Upon comparing this result with the provided answer choices, the correct choice is:

$\left(3\times8\right)^{5\times6}$

This choice correctly applies the power of a power rule, thereby validating the solution as correct.

In conclusion, the simplified form of the expression is $(3 \times 8)^{30}$, and the correct choice is option 4.