Q. 8.65 - Intermediate Algebra

Question

Simplify:

a. $x^{\frac{1}{6}}·x^{\frac{4}{3}}$ b.${\left(x^6\right)}^{\frac{4}{3}}$ c. $\frac{x^{{\displaystyle \frac{2}{3}}}}{x^{{\displaystyle \frac{5}{3}}}}$

Step-by-Step Solution

Answer

Part (a). The answer is $x^{\frac{3}{2}}$.

Part (b). The answer is $x^8$.

Part (c). The answer is $\frac{1}{x}$.

Part (a) Step 1. Simplifying

We know that if the base of the exponent is the same in the multiplication then we have to add the powers.

So,

$$\begin{gathered} =x^{\frac{1}{6}}·x^{\frac{4}{3}} \\ =x^{\frac{1}{6}+\frac{4}{3}} \\ =x^{\frac{1+8}{6}} \\ =x^{\frac{9}{6}} \\ =x^{\frac{3}{2}} \end{gathered}$$

Part (b) Step 1. Simplifying the exponent

We know that, 

${\left(a^b\right)}^c=a^{b·c}$

Similarly,

$$\begin{gathered} ={\left(x^6\right)}^{\frac{4}{3}}=x^{6·\frac{4}{3}} \\ =x^{2·4} \\ =x^8 \end{gathered}$$

Part (c) Step 1. Simplifying the exponent

We know that, $\frac{a^b}{a^c}=a^{b-c}$

Similarly,

$$\begin{gathered} =\frac{x^{{\displaystyle \frac{2}{3}}}}{x^{{\displaystyle \frac{5}{3}}}} \\ =x^{\frac{2}{3}-\frac{5}{3}} \\ =x^{\frac{2-5}{3}} \\ =x^{\frac{-3}{3}} \\ =x^{-1} \\ =\frac{1}{x} \end{gathered}$$