Q. 585. - Intermediate Algebra

Question

In the following exercises, simplify. Assume all variables are positive.

$\frac{x^{-\frac{1}{4}}{x}^{\frac{5}{4}}}{x^{-{\displaystyle \frac{3}{4}}}}$

Step-by-Step Solution

Answer

The simplified form of the given expression is $x^{\frac{7}{4}}$.

Step 1. Given information.

The given expression is $\frac{x^{-\frac{1}{4}}{x}^{\frac{5}{4}}}{x^{-{\displaystyle \frac{3}{4}}}}$

Step 2. Simplify the exponents.

Applying the Negative Exponent Property,

$\frac{x^{-\frac{1}{4}}{x}^{\frac{5}{4}}}{x^{-{\displaystyle \frac{3}{4}}}}=\frac{x^{\frac{3}{4}}{x}^{\frac{5}{4}}}{x^{{\displaystyle \frac{1}{4}}}}$

Now we will use the Product property,

$\frac{x^{\frac{3}{4}+{\displaystyle \frac{5}{4}}}}{x^{{\displaystyle \frac{1}{4}}}}=\frac{x^{{\displaystyle \frac{8}{4}}}}{x^{{\displaystyle \frac{1}{4}}}}$

Use Quotient Property,

$$\begin{gathered} \frac{x^{{\displaystyle \frac{8}{4}}}}{x^{{\displaystyle \frac{1}{4}}}}=x^{\frac{8}{4}-\frac{1}{4}} \\ =x^{\frac{7}{4}} \end{gathered}$$