Solve ((2×5)^a)^3: Nested Exponents with Base Multiplication

To solve this problem, let's follow these steps:

• Step 1: Identify the given information.
• Step 2: Apply the appropriate exponent rule.
• Step 3: Simplify and check the result against choices.

Now, let's work through each step:

Step 1: The given expression is $\left((2 \times 5)^a\right)^3$. Here, the base is $2 \times 5$, which is 10, but we don't need to compute it because we're focusing on exponent rules. The expression can be interpreted as $(10^a)^3$.

Step 2: We use the power of a power rule, which tells us that $(x^m)^n = x^{m \cdot n}$. In our case, $x = (2 \times 5)$, $m = a$, and $n = 3$. Applying the rule, we get: $((2 \times 5)^a)^3 = (2 \times 5)^{a \cdot 3} = (2 \times 5)^{3a}$

Step 3: The simplified expression is $(2 \times 5)^{3a}$. Comparing this with the given choices:

• Choice 1: $(2 \times 5)^{a+3}$ - Incorrect, as it adds exponents rather than multiplying them.
• Choice 2: $(2 \times 5)^{a-3}$ - Incorrect, as it subtracts exponents rather than multiplying them.
• Choice 3: $(2 \times 5)^{\frac{3}{a}}$ - Incorrect, as it divides exponents instead of multiplying them.
• Choice 4: $(2 \times 5)^{3a}$ - Correct, because it correctly applies the power of a power rule.

Therefore, the correct answer to the problem is $(2 \times 5)^{3a}$, which corresponds to choice 4.