Power of a Power: Applying the formula

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 this problem, we'll follow these steps:

• Step 1: Identify the given expression.

• Step 2: Apply the power of a power rule using exponent multiplication.

• Step 3: Confirm the result against the given choices.

Now, let's work through each step:

Step 1: Identify the Given Expression.
The given expression is $\left((15 \times 3)^{10}\right)^{10}$.

Step 2: Apply the Power of a Power Rule.
According to the exponent rule, $(a^m)^n = a^{m \times n}$, we multiply the exponents:

• The base is $15 \times 3$.

• The inner exponent is 10, and the outer exponent is 10.

• So, we apply the rule: $((15 \times 3)^{10})^{10} = (15 \times 3)^{10 \times 10}$.

• This simplifies to $(15 \times 3)^{100}$.

Step 3: Confirm the Result Against the Choices.
The expression simplifies to $(15 \times 3)^{100}$.

The correct choice from the options provided is:

$\left(15\times3\right)^{100}$

To solve this problem, we'll apply the power of a power rule on the expression $((4 \times 3)^3)^6$.

Here's how we proceed:

• Step 1: Identify the expression
  The expression given is $((4 \times 3)^3)^6$.

• Step 2: Apply the Power of a Power Rule
  According to the power of a power rule, $(a^m)^n = a^{m \times n}$.
  Therefore, $((4 \times 3)^3)^6 = (4 \times 3)^{3 \times 6}$.

• Step 3: Calculate the Exponent Product
  Multiply the exponents: $3 \times 6 = 18$.

• Step 4: Simplify the Expression
  Thus, we have $(4 \times 3)^{18}$.

Therefore, the simplified expression is $(4 \times 3)^{18}$.

Comparing with the choices provided:

• Choice 1: $(4 \times 3)^2$ - Incorrect.

• Choice 2: $(4 \times 3)^3$ - Incorrect.

• Choice 3: $(4 \times 3)^{18}$ - Correct.

• Choice 4: $(4 \times 3)^{-3}$ - Incorrect.

Thus, the correct answer is: $(4 \times 3)^{18}$, which corresponds to choice 3

To solve this problem, we'll follow these steps:

• Step 1: Analyze the given expression

• Step 2: Apply the appropriate exponent rule

• Step 3: Simplify to reach the final expression

Now, let's work through each step:

Step 1: Begin with the given expression, $\left(\left(7 \times 6\right)^5\right)^{10}$. Here, the inner expression $(7 \times 6)$ is raised to the fifth power, and this result is raised to the tenth power.

Step 2: Use the power of a power rule, which states that $(a^m)^n = a^{m \times n}$. Applying this rule to our expression, we identify $a = (7 \times 6)$, $m = 5$, and $n = 10$.

Step 3: Substitute these values into the formula:

$(7 \times 6)^{5 \times 10} = (7 \times 6)^{50}$

Therefore, the simplified expression is $(7 \times 6)^{50}$.

Upon comparison with the provided answer choices, choice 1 is correct:

• Choice 1: $(7 \times 6)^{50}$ - Correct, matches our simplified result.

• Choice 2: $(7 \times 6)^5$ - Incorrect, doesn't apply exponent rule.

• Choice 3: $(7 \times 6)^2$ - Incorrect, not relevant to problem scope.

• Choice 4: $(7 \times 6)^{15}$ - Incorrect, wrong application of formula.

Therefore, the final answer is $\left(7 \times 6\right)^{50}$.

To solve this problem, we'll follow these steps:

• Step 1: Identify the base and the exponents in the given expression.
• Step 2: Use the power of a power rule to simplify the expression.
• Step 3: Verify the solution against given answer choices.

Now, let's work through each step:

Step 1: The given expression is $((8 \times 9)^{11})^4$. Here, the base is $8 \times 9$, and the original exponent of the entire base is $11$. There is an outer exponent of $4$.

Step 2: Apply the power of a power rule, $(a^m)^n = a^{m \cdot n}$.
Thus, $((8 \times 9)^{11})^4 = (8 \times 9)^{11 \cdot 4}$.

Step 3: Perform the multiplication of exponents:
$11 \cdot 4 = 44$.
Therefore, $((8 \times 9)^{11})^4 = (8 \times 9)^{44}$.

Therefore, the solution to the problem is $(8 \times 9)^{44}$.

Now let's check the provided answer choices:

• Choice 1: $(8 \times 9)^{15}$ - Incorrect, as the operation is $(11 \times 4)$, not $(11 + 4)$.
• Choice 2: $(8 \times 9)^{44}$ - Correct, since $11 \times 4 = 44$.
• Choice 3: $(8 \times 9)^{\frac{4}{11}}$ - Incorrect, as this result is unrelated to multiplied exponents.
• Choice 4: $(8 \times 9)^7$ - Incorrect, as there is no reason to have a resulting exponent of $7$.

Therefore, the correct choice is Choice 2: $(8 \times 9)^{44}$.

To solve this problem, we'll follow these steps:

• Step 1: Identify the given expression.
• Step 2: Apply the "power of a power" rule for exponents.
• Step 3: Perform the necessary calculations to simplify the expression.

Now, let's work through each step:

Step 1: Identify the expression $\left(\left(10 \times 7\right)^8\right)^6$.
Step 2: Using the "power of a power" theorem, which states $(a^m)^n = a^{m \cdot n}$, we apply this to the expression.
Step 3: Inside our expression, $a = 10 \times 7$, $m = 8$, and $n = 6$. Thus, $\left(\left(10 \times 7\right)^8\right)^6$ becomes $\left(10 \times 7\right)^{8 \times 6}$.

Step 4: Calculate the new exponent: $8 \times 6 = 48$. Thus, the expression simplifies to $\left(10 \times 7\right)^{48}$.

Therefore, the solution to the problem is $\left(10 \times 7\right)^{48}$.

Now, let's consider the provided answer choices:

• Choice 1: $\left(10\times7\right)^2$ - Incorrect because this does not align with our calculation.
• Choice 2: $\left(10\times7\right)^{14}$ - Incorrect because the exponent is not calculated as $8 \times 6$.
• Choice 3: $\left(10\times7\right)^{\frac{2}{4}}$ - Incorrect because this exponent is not what results from $8 \times 6$.
• Choice 4: $\left(10\times7\right)^{48}$ - Correct, as it matches our simplified result.

We conclude that the correct solution is option 4.

We use the formula:

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

and therefore we obtain:

$(a^5)^7=a^{5\times7}=a^{35}$

We use the formula:

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

$(4^2)^3+(g^3)^4=4^{2\times3}+g^{3\times4}=4^6+g^{12}$

We use the power property of a negative exponent:

$a^{-n}=\frac{1}{a^n}$We will rewrite the fraction in parentheses as a negative power:

$\frac{1}{7}=7^{-1}$Let's return to the problem, where we had:

$\bigg( \big( \frac{1}{7}\big)^{-1}\bigg)^4=\big((7^{-1})^{-1} \big)^4$We continue and use the power property of an exponent raised to another exponent:

$(a^m)^n=a^{m\cdot n}$And we apply it in the problem:

$\big((7^{-1})^{-1} \big)^4 =(7^{-1\cdot-1})^4=(7^1)^4=7^{1\cdot4}=7^4$Therefore, the correct answer is option c

To solve this problem, we will follow these steps:

• Step 1: Apply the power of a power rule.

• Step 2: Simplify the resulting power.

Let's work through the process:

Step 1: Apply the power of a power rule, which states that $\left(a^m\right)^n = a^{m \times n}$.
We have $\left(\left(6\times5\right)^{-8}\right)^{-4}$. We can rewrite this using the power of a power rule:

$\left(\left(6\times5\right)^{-8}\right)^{-4} = \left(6\times5\right)^{-8 \times (-4)} = \left(6\times5\right)^{32}$

Step 2: By calculating the exponent: $-8 \times (-4) = 32$, we find the final simplified expression to be $\left(6\times5\right)^{32}$.

Therefore, the expression reduces to $\left(6\times5\right)^{32}$, which matches choice 2.

To solve the expression $\left(\left(7 \times 2\right)^3\right)^{-2}$, follow these steps:

• Step 1: Simplify the expression inside the first parentheses. $7 \times 2 = 14$
• Step 2: Apply the power of a power rule. $\left(14^3\right)^{-2}$ becomes $14^{3 \times (-2)}$
• Step 3: Calculate the exponent. $3 \times (-2) = -6$
• This gives us: $14^{-6}$
• The expression $\left(14^3\right)^{-2}$ simplifies to $\left(7 \times 2 \right)^{-6}$.

The expression simplifies to $\left(7 \times 2\right)^{-6}$.

Now, let's match this with the given choices:

• Choice 1: $\left(7 \times 2\right)^{-6}$ - This matches our result and is the correct answer.
• Choice 2: $\left(7 \times 2\right)^{\frac{-2}{3}}$ - Incorrect, since the power of a power rule wasn’t applied correctly.
• Choice 3: $\left(7 \times 2\right)^{-1}$ - Incorrect simplification of the given problem.
• Choice 4: $\left(7 \times 2\right)^1$ - Incorrect simplification showing misunderstanding of negative exponent rules.

Therefore, the correct answer is Choice 1: $\left(7 \times 2\right)^{-6}$.

To solve this problem, we need to simplify the expression $\left(\left(4 \times 8\right)^{-5}\right)^4$.

We'll follow these steps:

• Step 1: Understand the expression inside the parentheses $(4 \times 8)$ and calculate it.

• Step 2: Apply the power of a power property to simplify the expression.

Step 1: Calculate the expression inside the parentheses.
We have $4 \times 8 = 32$, so the expression becomes $\left(32^{-5}\right)^4$.

Step 2: Apply the power of a power property.
This property states that $(a^m)^n = a^{m \cdot n}$. Here, $m = -5$ and $n = 4$, so:

$(32^{-5})^4 = 32^{-5 \times 4} = 32^{-20}$.

Therefore, the simplified expression is $\left(4 \times 8\right)^{-20}$.

Hence, the correct answer choice is:

• Choice 4: $\left(4\times8\right)^{-20}$

All other choices result from errors in applying the exponent rules or miscalculating intermediate steps:

• Choice 1: Misapplies the exponent rules, yielding $-9$ instead of $-20$.

• Choice 2: Incorrectly calculates the expression, resulting in $-1$.

• Choice 3: Incorrect fractional exponent interpretation does not apply here.

To solve this problem, we'll follow these steps:

• Step 1: Identify the expression given: $\left(\left(9 \times 3\right)^{-4}\right)^{-6}$
• Step 2: Apply the power of a power rule
• Step 3: Simplify the expression

Now, let's work through each step:

Step 1: The given expression is $\left(\left(9 \times 3\right)^{-4}\right)^{-6}$.

Step 2: We'll apply the power of a power rule which states that $\left(a^m\right)^n = a^{m \cdot n}$. In our case:

• The base $a$ is $9 \times 3$
• The inner exponent $m$ is $-4$
• The outer exponent $n$ is $-6$

By applying the formula, we get:

Step 3: Calculate the exponent:

Thus, the expression simplifies to:

Therefore, the solution to the problem is $\left(9 \times 3\right)^{24}$.

Examining the multiple-choice options, the correct choice is:

• Choice 2: $\left(9\times3\right)^{24}$

Thus, Choice 2 is the correct answer, aligning with our calculated solution.

To solve the problem, we'll apply the exponent rule that states $\left(a^m\right)^n = a^{m \times n}$. Here’s how we proceed:

• Step 1: Recognize that the expression inside is $\left((10 \times 3)^{-4}\right)$, which is then raised to the 7th power.

• Step 2: Use the Power of a Power Rule: $\left(a^m\right)^n = a^{m \times n}$.

• Step 3: Applying this formula to our expression $\left((10 \times 3)^{-4}\right)^7$, results in $(10 \times 3)^{-4 \times 7}$.

• Step 4: Compute the multiplication in the exponent: $-4 \times 7 = -28$.

Therefore, $\left(\left(10\times3\right)^{-4}\right)^7 = (10 \times 3)^{-28}$.

Now, we need to compare our solution with the given choices:

• Choice 1: $(10 \times 3)^3$.

• Choice 2: $(10 \times 3)^{-11}$.

• Choice 3: $(10 \times 3)^{-28}$.

• Choice 4: $(10 \times 3)^{-\frac{7}{4}}$.

The correct choice is Choice 3: $(10 \times 3)^{-28}$, as this matches our simplified expression.