Power of a Power: Calculating powers with negative exponents

To solve this problem, we'll apply the power of a power rule for exponents. The problem is to simplify the expression $\left(\left(8\times6\right)^{-7}\right)^{-8}$.

• Step 1: Understand the Power of a Power Rule

The power of a power rule states that $(a^m)^n = a^{m \times n}$.

• Step 2: Apply the Rule

Here, the base of the entire expression is $8 \times 6$, the first exponent is $-7$, and the second exponent is $-8$. According to the rule, we multiply the exponents:

$(8 \times 6)^{-7 \times -8}$

• Step 3: Simplify the Exponent Calculation

Calculate the multiplication of the exponents:

$-7 \times -8 = 56$

This results in the expression:

$(8 \times 6)^{56}$

Considering the given choices, carefully cross-check against our simplified expression:

• Choice 1: $\left(8\times6\right)^{-7-8}$ is incorrect - it uses addition instead of multiplication.

• Choice 2: $\left(8\times6\right)^{\frac{-8}{-7}}$ is incorrect - it uses division instead.

• Choice 3: $\left(8\times6\right)^{-7+8}$ is incorrect - it uses addition as well.

• Choice 4: $\left(8\times6\right)^{-7\times-8}$ is our correct transformation before final simplification.

After calculating, the expression $\left(8\times6\right)^{-7\times-8} = (8 \times 6)^{56}$. The corresponding expression reflects Choice 4 before final simplification.

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

• Step 1: Identify the expression given and simplify the base.
• Step 2: Apply the Power of a Power Rule for exponents.
• Step 3: Match the result with the provided answer choices.

Let's work through each step:

Step 1: The problem gives us the expression $\left(\left(2\times4\right)^{-2}\right)^4$. First, simplify the base: $2 \times 4$ equals 8, so the expression becomes $\left(8^{-2}\right)^4$.

Step 2: Apply the Power of a Power Rule, which states: $(a^m)^n = a^{m \times n}$. Here, $a = 8$, $m = -2$, and $n = 4$. Calculate $m \times n = -2 \times 4 = -8$.

Therefore, $\left(8^{-2}\right)^4$ simplifies to $8^{-8}$, which can be expressed back in terms of the original base $(2 \times 4)$. So we write it as $\left(2 \times 4\right)^{-8}$.

Step 3: Check the given choices:

• Choice 1: $\left(2\times4\right)^{-2+4}$ represents an exponent of 2; incorrect.
• Choice 2: $\left(2\times4\right)^{\frac{4}{-2}}$ simplifies to -2; incorrect.
• Choice 3: $\left(2\times4\right)^{-2-4}$ simplifies to -6; incorrect.
• Choice 4: $\left(2\times4\right)^{-2\times4} = \left(2\times4\right)^{-8}$; correct.

Thus, the correct choice is Choice 4: $\left(2\times4\right)^{-2\times4}$.

To simplify the expression $\left(\left(3\times5\right)^{-3}\right)^{-6}$, we apply exponent rules, specifically the power of a power rule.

Here's a step-by-step solution:

• Step 1: Identify the structure of the expression: we have an outer exponent $-6$ and an inner exponent $-3$ applied to the base $(3 \times 5)$.

• Step 2: Apply the power of a power rule, which states $(a^m)^n = a^{m \cdot n}$. This combines the exponents being multiplied.

• Step 3: Calculate the multiplication of the exponents: $-3 \times -6$. This yields $18$ since multiplying two negative numbers results in a positive number.

• Step 4: Substitute back into the expression: $\left(3 \times 5\right)^{18}$.

Thus, the transformation of the original expression results in the new expression:

$\left(3\times5\right)^{-3\times-6}$

Comparing with the provided choices, we see:

• Choice 1: $\left(3\times5\right)^{-3\times-6}$ - This matches our solution.

• Choices 2, 3, and 4 do not match the derived steps based on multiplying exponents.

Thus, the correct answer is choice 1: $\left(3\times5\right)^{-3\times-6}$.

To solve this problem, we'll simplify the expression $\left(\left(6\times2\right)^4\right)^{-5}$ using exponent rules.

Here's a step-by-step breakdown of the solution:

• Step 1: Identify the form
  The expression is $\left((6 \times 2)^4\right)^{-5}$. This is a case of the power of a power: $(a^m)^n$, which can be rewritten as $a^{m \times n}$.
• Step 2: Apply the power of a power rule
  Apply the rule: $\left((6 \times 2)^4\right)^{-5} = (6 \times 2)^{4 \times (-5)}$.
• Step 3: Simplify the exponent
  Calculate the exponent multiplication: $4 \times (-5) = -20$. Thus, the expression simplifies to $(6 \times 2)^{-20}$.

The resulting expression matches the format of choice 3: $\left(6 \times 2\right)^{4\times-5}$.

Therefore, the correct choice is Choice 3, $\left(6\times2\right)^{4\times-5}$.

To simplify the expression $\left(\left(7\times4\right)^{-6}\right)^5$, follow these steps, checking against the choices provided:

Step 1: Apply the power of a power rule.

• The expression inside the parentheses $7\times4$ acts as a single term $a$.

• Therefore, by applying $(a^m)^n = a^{m \times n}$, we simplify:

$\left(\left(7\times4\right)^{-6}\right)^5 = \left(7\times4\right)^{-6 \times 5}$

Step 2: Multiply the exponents.

• Calculate $-6 \times 5 = -30$.

• Hence, the expression simplifies to $\left(7\times4\right)^{-30}$.

Conclusion:

The correct simplified form of the expression is $\left(7\times4\right)^{-6\times5}$, aligning with choice 2 and your provided correct answer.

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.

To solve the problem, let us simplify the expression $\left(\left(5\times3\right)^4\right)^{-3}$.

First, recognize that the expression inside the parentheses, $5 \times 3$, can be multiplied to give us 15. However, we'll focus on exponent rules directly.

• Step 1: Apply the Power of a Power Rule.
  Using the formula $(a^m)^n = a^{m \times n}$, simplification gives: $\left(\left(5\times3\right)^4\right)^{-3} = \left( (5 \times 3)^{4 \times -3} \right) = \left(5\times3\right)^{-12}.$
• Step 2: Apply the Negative Exponent Rule.
  Using $a^{-n} = \frac{1}{a^n}$, the expression becomes: $\left(5\times3\right)^{-12} = \frac{1}{\left(5\times3\right)^{12}}.$

Therefore, the solution to the given expression is $\frac{1}{\left(5\times3\right)^{12}}$.

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

• Choice 1: $\frac{1}{\left(5\times3\right)^{12}}$ - This matches our solution.
• Choice 2: $(5\times3)^{12}$ - Incorrect, doesn't account for the negative exponent.
• Choice 3: $\frac{1}{\left(5\times3\right)^{-1}}$ - Incorrect power and doesn't represent the full expression.
• Choice 4: $\frac{1}{\left(5\times3\right)^1}$ - Incorrect power and doesn't reflect the original problem.

Thus, the correct choice is Choice 1: $\frac{1}{\left(5\times3\right)^{12}}$.

To solve this problem, we must simplify the expression $\left(\left(7 \times 6\right)^{-5}\right)^3$.

We'll follow these steps:

• Step 1: Apply the power of a power rule.
• Step 2: Simplify the expression into a single exponent.
• Step 3: Convert the negative exponent to a fraction form.

Now, let's work through each step:

Step 1: The expression $\left(7 \times 6\right)^{-5}$ is raised to the power 3. By the power of a power rule, we multiply the exponents:

Step 2: This simplifies the expression to $(7 \times 6)^{-15}$.

Step 3: Since we have a negative exponent, we convert it to a fraction:

Therefore, the simplified expression is:

Comparing this result with the given choices, the correct answer is:

The other choices are incorrect because they either have the wrong exponent or incorrectly handle the negative exponent.

Thus, the correct answer to the problem is $\frac{1}{\left(7 \times 6\right)^{15}}$.

To solve this expression, we will follow these steps using the rules of exponents:

• Step 1: Apply the power of a power rule.
• Step 2: Use the negative exponent rule to express the result as a fraction.

Now, let's apply each step:

Step 1: Apply the power of a power rule
Given: $\left(\left(8\times4\right)^{-7}\right)^6$.
According to the power of a power rule, $(a^m)^n = a^{m \cdot n}$.
So, $\left(\left(8\times4\right)^{-7}\right)^6 = \left(8\times4\right)^{-7 \times 6} = \left(8\times4\right)^{-42}$.

Step 2: Use the negative exponent rule
Now, apply the negative exponent rule: $a^{-m} = \frac{1}{a^m}$.
Thus, $\left(8\times4\right)^{-42} = \frac{1}{\left(8\times4\right)^{42}}$.

The simplified expression is $\frac{1}{\left(8\times4\right)^{42}}$.

Now, let's determine which of the provided answer choices is correct:
- Choice 1: $\frac{1}{\left(8\times4\right)^{-42}}$ is incorrect because the exponent should not be negative.
- Choice 2: $\frac{1}{\left(8\times4\right)^{42}}$ is correct as it matches our solution.
- Choice 3: $\frac{1}{\left(8\times4\right)^{-1}}$ is incorrect because it does not match our calculated exponent.
- Choice 4: $\frac{1}{\left(8\times4\right)^1}$ is incorrect as the exponent is too small.

Therefore, the correct answer is $\frac{1}{\left(8\times4\right)^{42}}$, which corresponds to Choice 2.