Question

chris tried to rewrite the expression \\(\left(4^{-2} \cdot 4^{-3}\
ight)^{3}\\).
\\(\

$$\begin{align*}\\left(4^{-2} \\cdot 4^{-3}\ ight)^{3}\\\\&= \\left(4^{-5}\ ight)^{3} & \\text{step 1}\\\\&= 4^{-2} & \\text{step 2}\\\\&= \\frac{1}{4^{2}} & \\text{step 3}\\end{align*}$$

did chris make a mistake? if so, in which step?
choose 1 answer:
a chris did not make a mistake.
b chris made a mistake in step 1.
c chris made a mistake in step 2.
d chris made a mistake in step 3.

Explanation:

Step1: Verify Step 1 of Chris's work

When multiplying exponents with the same base, add exponents: $a^m \cdot a^n = a^{m+n}$.
$4^{-2} \cdot 4^{-3} = 4^{-2 + (-3)} = 4^{-5}$. This step is correct.

Step2: Verify Step 2 of Chris's work

When raising a power to a power, multiply exponents: $(a^m)^n = a^{m \cdot n}$.
$(4^{-5})^3 = 4^{-5 \times 3} = 4^{-15}$. Chris incorrectly calculated this as $4^{-2}$, so this step has a mistake.

Step3: Verify Step 3 of Chris's work

A negative exponent rule: $a^{-n} = \frac{1}{a^n}$. If the exponent were $-2$, $\frac{1}{4^2}$ would be correct, but the error occurred earlier.

Answer:

C. Chris made a mistake in Step 2.