Power of a Power: Presenting powers with negative exponents as fractions

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:

$\left((7 \times 6)^{-5}\right)^3 = (7 \times 6)^{-5 \times 3} = (7 \times 6)^{-15}$

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:

$(7 \times 6)^{-15} = \frac{1}{(7 \times 6)^{15}}$

Therefore, the simplified expression is:

$\frac{1}{\left(7 \times 6\right)^{15}}$

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

- Choice 3: $\frac{1}{\left(7 \times 6\right)^{15}}$

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.