Problem 60 - College Algebra Essentials

Question

Simplify each exponential expression. $$\left(\frac{3 x^{4}}{y}\right)^{-3}$$

Step-by-Step Solution

Answer

\(\frac{y^3}{27x^{12}}\)

Step 1: Apply the Negative Exponent Rule

The negative exponent rule states that \(a^{-n} = \frac{1}{a^{n}}\). Applying this to our expression gives: \(\left(\frac{3 x^{4}}{y}\right)^{-3} = \frac{1}{\left(\frac{3 x^{4}}{y}\right)^{3}}\)

Step 2: Apply the Power of a Quotient Rule

This rule states that \(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\). In this step apply it: \(\frac{1}{\left(\frac{3 x^{4}}{y}\right)^{3}} = \frac{1}{\left(3^{3} x^{4*3} y^{-1*3}\right)}\) which simplifies to \(\frac{1}{27x^{12}y^{-3}}\)

Step 3: Apply the Negative Exponent Rule Again

The negative exponent rule is applied again to manage \(y^{-3}\): \(\frac{1}{27x^{12}y^{-3}} = \frac{1}{27x^{12}} * \frac{1}{y^{-3}} = \frac{y^3}{27x^{12}}\)

Step 4: Final Simplified Expression

No more simplification is possible, hence the final answer is: \(\frac{y^3}{27x^{12}}\)