Question

chris tried to rewrite the expression $(4^{-2}cdot4^{-3})^{3}$. step 1: $(4^{-2}cdot4^{-3})^{3}=(4^{-5})^{3}$ step 2: $(4^{-5})^{3}=4^{-2}$ step 3: $4^{-2}=\frac{1}{4^{2}}$ wheres the mistake? did chris make a mistake? if so, in which step? chris did not make a mistake. chris made a mistake in step 1. chris made a mistake in step 2. chris made a mistake in step 3.

Explanation:

Step1: Apply product - of - powers rule

When multiplying two numbers with the same base \(a^m\cdot a^n=a^{m + n}\), so \((4^{-2}\cdot4^{-3})^3=(4^{-2+( - 3)})^3=(4^{-5})^3\). Chris did this step correctly.

Step2: Apply power - of - a - power rule

When raising a power to a power \((a^m)^n=a^{m\times n}\), so \((4^{-5})^3 = 4^{-5\times3}=4^{-15}\), not \(4^{-2}\). Chris made a mistake in this step.

Step3: Apply negative - exponent rule

The negative - exponent rule is \(a^{-n}=\frac{1}{a^n}\), if the previous step was correct, \(4^{-15}=\frac{1}{4^{15}}\), but since step 2 was wrong, this step is also wrong based on the wrong result of step 2.

Answer:

Chris made a mistake in Step 2.