Power of a Product: Variable in the exponent of the power

To solve the problem of expanding $(2a)^{y+5}$, we'll use the Power of a Product Rule.

• Step 1: Identify the base and the exponent. Here, the base is $2a$, and the exponent is $y+5$.
• Step 2: Apply the Power of a Product Rule, which states that $(ab)^n = a^n \times b^n$. In this case, apply it to the base $2a$.
• Step 3: Expand the expression: $(2a)^{y+5} = 2^{y+5} \times a^{y+5}$.

By applying the rule, we separate the exponential expression into two parts, one for each component of the base:

$(2a)^{y+5} = 2^{y+5} \times a^{y+5}$

This result shows that both $2$ and $a$ are individually raised to the power of $y+5$. The application of the product rule ensures that each base component is treated equally within the exponentiation.

Therefore, the expanded form of the expression is $2^{y+5} \times a^{y+5}$, which corresponds to answer choice 4.

To solve this problem, we will use the rules for handling exponents:

• Step 1: Start with the original expression $(4 \times a)^{-x}$.

• Step 2: Apply the Power of a Product rule: $(4 \times a)^{-x} = 4^{-x} \times a^{-x}$.

• Step 3: Apply the Negative Exponent rule for each factor: $4^{-x} = \frac{1}{4^x}$ and $a^{-x} = \frac{1}{a^x}$.

• Step 4: Combine the results of Step 3, resulting in: $\frac{1}{4^x \times a^x}$.

The equivalent expression for $(4 \times a)^{-x}$ is $\frac{1}{4^x \times a^x}$.

By comparing this with the given choices, the correct answer choice is:

$\frac{1}{4^x \times a^x}$

To solve this problem, we'll employ the following strategy:

• Step 1: Apply the power of a product rule: $\left(a \times x\right)^{-t} = a^{-t} \times x^{-t}$.
• Step 2: Apply the negative exponent rule: $a^{-t} = \frac{1}{a^t}$ and $x^{-t} = \frac{1}{x^t}$.

Here's how we do it step by step:
Step 1: We have $\left(a \times x\right)^{-t}$. By the power of a product rule, this rewrites as $a^{-t} \times x^{-t}$.
Step 2: Apply the negative exponent rule to each part:
$a^{-t} = \frac{1}{a^t}$ and $x^{-t} = \frac{1}{x^t}$.
Therefore, $a^{-t} \times x^{-t} = \frac{1}{a^t} \times \frac{1}{x^t} = \frac{1}{a^t \times x^t}$.

Thus, the solution to the equation $\left(a \times x\right)^{-t}$ is $\frac{1}{a^t \times x^t}$.