Simplify the Nested Exponent Expression: ((x×b)⁵)⁸

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

• Step 1: Identify the mathematical problem and the rule needed.
• Step 2: Apply the power of a power rule to simplify the expression.
• Step 3: Evaluate our simplification against the provided answer choices.

Now, let's work through each step:

Step 1: The problem involves simplifying the expression $\left(\left(x \times b\right)^5\right)^8$, which is a power of a power. Our goal is to simplify it to a single power.

Step 2: According to the power of a power rule, $(a^m)^n = a^{m \times n}$. In this case, the base is $(x \times b)$ raised to the 5th power and that entire expression is further raised to the 8th power.

Applying the power of a power rule gives us:

$\left(\left(x \times b\right)^5\right)^8 = \left(x \times b\right)^{5 \times 8} = \left(x \times b\right)^{40}$

Step 3: Compare this to the provided answer choices:

• Choice 1: $\left(x \times b\right)^{5+8}$ – This is incorrect as it adds the exponents instead of multiplying them.
• Choice 2: $\left(x \times b\right)^{5 \times 8}$ – This is correct as it applies the rule correctly to yield $(x \times b)^{40}$.
• Choice 3: $\left(x \times b\right)^{\frac{8}{5}}$ – This is incorrect and irrelevant to the power of a power rule.
• Choice 4: $\left(x \times b\right)^{5-8}$ – This is incorrect as it subtracts exponents, which is not applicable here.

Therefore, after careful analysis, the solution to the problem is the expression represented by Choice 2: $\left(x \times b\right)^{40}$.