Problem 59 - College Algebra Essentials

Question

Simplify each exponential expression. $$\left(\frac{5 x^{3}}{y}\right)^{-2}$$

Step-by-Step Solution

Answer

The simplified form of \(\left(\frac{5 x^{3}}{y}\right)^{-2}\) is \(\frac{y^2}{25x^6}\).

Step 1: Distribute the Exponent

Start by distributing the exponent -2 to each part of the fraction in the base. This results in \((5^{-2})(x^{3*-2})/(y^{-2})\).

Step 2: Calculate the Negative Exponents

Now, remember that a negative exponent simply inverts the base, i.e, \(a^{-n} = \frac{1}{a^n}\). With this rule, replace each part with negative exponents, getting \(\frac{1}{5^2}*\frac{1}{x^{3*2}}*y^2\).

Step 3: Simplify the Expression

Now calculate the remaining exponents and simplify the result into simplest terms. This gives: \(\frac{1}{25}*\frac{1}{x^{6}}*y^2\), which simplifies to \(\frac{y^2}{25x^6}\).