Solve (5×8)^a^b: Evaluating Nested Power Expressions

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

• Step 1: Identify the given expression.
• Step 2: Apply the power of a power rule.
• Step 3: Simplify the expression to find the correct answer.

Now, let's work through each step:

Step 1: The initial mathematical expression given is $\left(\left(5\times8\right)^a\right)^b$.
Step 2: We apply the power of a power rule, which states that $\left(x^m\right)^n = x^{m \times n}$.
Step 3: Applying this rule, we get:

$\left(\left(5\times8\right)^a\right)^b = \left(5\times8\right)^{a \times b}$

Thus, the simplified expression is $\left(5\times8\right)^{a \times b}$.

Comparing this with the choices given:

• Choice 1: $(5\times8)^{a-b}$ - Incorrect, as it uses subtraction instead of multiplication.
• Choice 2: $(5\times8)^{a+b}$ - Incorrect, as it uses addition instead of multiplication.
• Choice 3: $(5\times8)^{a \times b}$ - Correct, as it uses multiplication, as derived.
• Choice 4: $(5\times8)^{\frac{a}{b}}$ - Incorrect, as it uses division instead of multiplication.

Therefore, the correct answer is $\left(5\times8\right)^{a\times b}$, which corresponds to choice 3.