Solve (4x)^5)^3: Evaluating Nested Exponent Expression

To solve this problem, we will apply the power of a power rule for exponents, which states that $\left(a^m\right)^n = a^{m \times n}$.

**Step-by-step Solution:**

• Step 1: Identify the given information.

  • The expression is $\left(\left(4\times x\right)^5\right)^3$.

  • The base is $4\times x$, the first exponent is 5, and the second exponent is 3.

• Step 2: Apply the power of a power rule.

  • According to the rule: $\left((4 \times x)^5\right)^3 = (4 \times x)^{5 \times 3}$.

  • Multiply the exponents: $5 \times 3 = 15$.

  • Thus, the expression simplifies to $(4 \times x)^{15}$.

Therefore, the simplified expression is $(4 \times x)^{15}$.

Choice Analysis:

• The correct choice is:

  $\left(4\times x\right)^{5\times3}$

  , which correctly applies the power of a power rule.

• Incorrect choices:

  • : $(4\times x)^{5+3}$ – Incorrect, it adds exponents instead of multiplying.

  • : $(4\times x)^{\frac{5}{3}}$ – Incorrect, it uses division but should multiply exponents.

  • : $(4\times x)^{5-3}$ – Incorrect, it subtracts exponents instead of multiplying.