Question

Simplify:

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

Short answer

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

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

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

Step-by-step solution

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,

localid="1644693096053" $$\begin{gathered} =\frac{x^{{\displaystyle \frac{2}{3}}}}{x^{{\displaystyle \frac{5}{3}}}} \\ =x^{\frac{2}{3}-\frac{5}{3}} \\ =x^{\frac{2-5}{3}} \\ =x^{\frac{-3}{3}} \\ =x^{-1} \\ =\frac{1}{x} \end{gathered}$$