Solve Nested Powers: Simplifying ((5³)⁴)⁶ Using Exponent Laws

To solve this problem, we'll use the power of a power property of exponents. This states that when raising a power to another power, we multiply the exponents.

• Step 1: Apply the power of a power rule to the inner expression $(5^3)^4$.
• Step 2: Simplify the result from Step 1 using the power of a power rule again with the outer power.

Let's break it down step-by-step:

Step 1:
We start with the innermost expression: $(5^3)^4$. According to the power of a power property, $(a^m)^n = a^{m \cdot n}$, so:

$(5^3)^4 = 5^{3 \cdot 4} = 5^{12}$

Step 2:
Now take the result from step 1, $5^{12}$, and raise it to the 6th power:

$(5^{12})^6 = 5^{12 \cdot 6} = 5^{72}$

Therefore, the simplified expression is $\boxed{5^{72}}$.

Matching this result with the choices provided, the correct answer is:

$5^{72}$

Choice (5^{72}) is correct.

When evaluating the incorrect choices:

• Choice (5^{13}): Incorrect because the exponents are not simply added.
• Choice (5^{18}): Incorrect because the exponents are not multiplied accurately.
• Choice (5^{65}): Incorrect as it's derived from a miscalculation.

I am confident that the solution is correct, as it follows directly from applying the correct exponent rules thoroughly and logically.