Questions: Simplify the expression so that no negative exponents appear in the final result. Assume all variables represent nonzero numbers. (x^-3 y^6)^-4 Select one: a. x^-7/y^2 b. 1/(x^12 y^24) c. x^12/y^24

Simplify the expression so that no negative exponents appear in the final result. Assume all variables represent nonzero numbers.

(x^-3 y^6)^-4

Select one:
a. x^-7/y^2
b. 1/(x^12 y^24)
c. x^12/y^24

Transcript text: \# 4.4 Practice for Class Time left 2:18:48 Simplify the expression so that no negative exponents appear in the final result. Assume all variables represent nonzero numbers. \[ \left(x^{-3} y^{6}\right)^{-4} \] Select one: a. $\frac{x^{-7}}{y^{2}}$ b. $\frac{1}{x^{12} y^{24}}$ c. $\frac{x^{12}}{y^{24}}$

Solution

Solution Steps

To simplify the expression $\left(x^{-3} y^{6}\right)^{-4}$ so that no negative exponents appear in the final result, we need to apply the power of a power rule $(a^m)^n = a^{m \cdot n}$. This means we will multiply the exponents inside the parentheses by $-4$.

Solution Approach

Apply the power of a power rule to each term inside the parentheses.
Simplify the resulting expression to ensure no negative exponents remain.

Step 1: Apply the Power of a Power Rule

Given the expression $\left(x^{-3} y^{6}\right)^{-4}$, we apply the power of a power rule $(a^m)^n = a^{m \cdot n}$ to each term inside the parentheses: \[ (x^{-3})^{-4} \quad \text{and} \quad (y^{6})^{-4} \]

Step 2: Simplify Each Term

Simplify each term by multiplying the exponents: \[ (x^{-3})^{-4} = x^{-3 \cdot -4} = x^{12} \] \[ (y^{6})^{-4} = y^{6 \cdot -4} = y^{-24} \]

Step 3: Combine the Results

Combine the simplified terms: \[ x^{12} \cdot y^{-24} \]

Step 4: Eliminate Negative Exponents

To ensure no negative exponents appear in the final result, rewrite $y^{-24}$ as $\frac{1}{y^{24}}$: \[ x^{12} \cdot \frac{1}{y^{24}} = \frac{x^{12}}{y^{24}} \]