Q. 583. - Intermediate Algebra

Question

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

(a) ${216}^{-\frac{1}{4}}$

(b) $-{49}^{\frac{3}{2}}$

Step-by-Step Solution

Answer

The simplified forms are,

(a) ${\left(\frac{1}{216}\right)}^{\frac{1}{4}}$

(b) $343i$

Part (a) Step 1. Simplify the given number.

Applying the Negative Exponent Property,

$$\begin{gathered} {216}^{-\frac{1}{4}}=\frac{1}{{216}^{\frac{1}{4}}} \\ ={\left(\frac{1}{216}\right)}^{\frac{1}{4}} \end{gathered}$$

Part (b) Step 1. Simplify the given number.

Applying the Power Property

$$\begin{gathered} -{49}^{\frac{3}{2}}={\left(-{49}^3\right)}^{\frac{1}{2}} \\ ={\left(-117649\right)}^{{}^{\frac{1}{2}}} \\ =\sqrt{-117649} \\ =\sqrt{-1}\sqrt{-117649} \\ =343i \end{gathered}$$