Power of a Power: Inverse formula

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

• Step 1: Identify the structure and components of the initial expression
• Step 2: Apply the inverse power of a power rule
• Step 3: Find the matching choice among the given options

Now, let's work through each step:

Step 1: The given expression is $\left(4 \times a\right)^{2b}$. This indicates that the base $\left(4 \times a\right)$ is raised to the power of $2b$.

Step 2: By the inverse power of a power property, we rewrite $\left(4 \times a\right)^{2b}$ in a way that exposes it as a power raised to a power. The expression $(x^{m \cdot n})$ equates to $(x^m)^n$. Hence, we can rewrite $\left(4 \times a\right)^{2b}$ as $\left(\left(4 \times a\right)^2\right)^b$.

Step 3: The correct answer from the provided choices matches choice 1: $\left(\left(4 \times a\right)^2\right)^b$.

Therefore, applying the inverse power of a power rule, the expression $\left(4 \times a\right)^{2b}$ becomes $\left(\left(4 \times a\right)^2\right)^b$.

To solve this problem, we will rewrite the expression $(15)^{xy}$ using the rules of exponents.

• Step 1: Understand that $(15)^{xy}$ can be rewritten using the power of a power rule.
• Step 2: Apply the exponent rule $(a^m)^n = a^{m \times n}$. We know $(15^x)^y = (15)^{x \times y}$ and $(15^y)^x = (15)^{y \times x}$, both equivalent to $(15)^{xy}$.
• Step 3: Analyze each choice:

Choice 1: $(15^y)^x$ is equivalent to $(15)^{xy}$ since applying the rule gives us $(15^y)^x = (15)^{y \times x} = (15)^{xy}$.
Choice 2: $(15^x)^y$ is also equivalent to $(15)^{xy}$ because applying the rule provides $(15^x)^y = (15)^{x \times y} = (15)^{xy}$.
Choice 3: $15^x \times 15^y$ results in $15^{x+y}$, which is not equivalent to $(15)^{xy}$ as it uses the product of powers rule.\
Choice 4: Both $(15^y)^x$ and $(15^x)^y$ are correct based on the rules involved.

Based on the analysis, choice 4 (a'+b' are correct) is the correct answer.
Both $(15^y)^x$ and $(15^x)^y$ are equivalent representations of $(15)^{xy}$.

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

• Step 1: Understand the expression $10^{3x}$.
• Step 2: Apply the power of a power rule to rewrite it.
• Step 3: Identify the correct equivalent expression from the options.

Now, let's work through each step:
Step 1: The expression given is $10^{3x}$, which involves a base of 10 and a combination of numerical and variable exponents, specifically $3x$.
Step 2: To rewrite this expression, we use the power of a power rule for exponents, which states $(a^m)^n = a^{m \cdot n}$. In our case, we want to reverse this process: we express $10^{3x}$ as $(10^3)^x$. Here, by viewing $3x$ as the product of $3$ and $x$, we can apply the rule effectively.
Step 3: We now compare our converted expression $(10^3)^x$ with the provided answer choices. The correct rewritten form is:
- Choice 3: $\left(10^3\right)^x$
Therefore, the solution to the problem is $\left(10^3\right)^x$. This matches the correct answer provided, validating our analysis and application of the power of a power rule.

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

• Step 1: Identify and restate the given expression.
• Step 2: Decide on the exponent technique to use.
• Step 3: Rewrite the expression using the identified rule.

Now, let's work through each step:
Step 1: The problem gives us the expression $(10x)^{8x}$.
Step 2: We will apply the power of a power rule for exponents, which states that if you have $(a^m)^n$, it equals $a^{m \cdot n}$.
Step 3: We need to express $8x$ as a product of two numbers. Let's write $8x$ as $(2x) \cdot 4$. Hence, using the power of a power rule, we can transform this into $((10x)^{2x})^4$.

Therefore, the solution to the problem is $(10x)^{8x} = ((10x)^{2x})^4$.

Upon examining the given answer choices, we find that choice 1: $\left(\left(10x\right)^{2x}\right)^4$ matches our solution. The other choices do not represent the original expression correctly using the power of a power rule.

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

• Step 1: Identify the given expression as $(x \times y)^{4a}$.
• Step 2: Apply the power of a power rule $(a^m)^n = a^{m \times n}$.
• Step 3: Transform the expression using these mathematical rules.

Now, let's work through each step:

Step 1: The problem provides us with the expression $(x \times y)^{4a}$. This expression involves a product raised to a power $4a$.

Step 2: We aim to express this using the idea of a power raised to another power. According to this rule, we can interpret $(x \times y)^{4a}$ as $((x \times y)^4)^a$. The rule applied here is $(a \times b)^n = a^n \times b^n$ in reverse, leading to $(a^m)^n = a^{m \times n}$ understanding for breaking down.

Step 3: Thus, $(x \times y)^{4a}$ becomes $((x \times y)^4)^a$.

After analyzing the answer choices:

• Choice 1, $\left(x \times y\right)^4 \times \left(x \times y\right)^a$, does not use the power of a power rule, it is a product.
• Choice 2, $\left((x \times y)^4\right)^a$, correctly represents the power of a power rule.
• Choice 3, $\frac{\left(x \times y\right)^4}{\left(x \times y\right)^a}$, represents division of powers, not the intended structure.
• Choice 4 cannot be correct as it lists combinations that do not follow from the given rules.

Therefore, the correct solution to the problem is $\left((x \times y)^4\right)^a$, which is answer choice 2.