Solve (2×3)²)⁵: Evaluating Nested Exponential Expressions

To solve the problem, we will simplify the expression $\left(\left(2 \times 3\right)^2\right)^5$ using the power of a power exponent rule. Follow these steps:

• Step 1: Identify the form of the expression. The given expression is $\left(\left(2 \times 3\right)^2\right)^5$.
• Step 2: Apply the power of a power rule, which states that $(a^m)^n = a^{m \times n}$.
• Step 3: Here, the base is $2 \times 3$, the first exponent ($m$) is 2, and the second exponent ($n$) is 5.
• Step 4: Multiply the exponents: $2 \times 5 = 10$.

Therefore, the expression simplifies to $\left(2 \times 3\right)^{10}$. However, for the purpose of matching the form requested, it can be expressed as $\left(2 \times 3\right)^{2 \times 5}$.

Next, we evaluate the given choices:

• Choice 1: $\left(2 \times 3\right)^{2+5}$ — This incorrectly adds the exponents instead of multiplying them.
• Choice 2: $\left(2 \times 3\right)^{5-2}$ — This incorrectly subtracts the exponents.
• Choice 3: $\left(2 \times 3\right)^{2\times5}$ — This correctly multiplies the exponents, which we found is the right simplification.
• Choice 4: $\left(2 \times 3\right)^{\frac{5}{2}}$ — This introduces division of exponents, which is not applicable here.

The correct choice is Choice 3: $\left(2 \times 3\right)^{2\times5}$.