Q. 8.66 - Intermediate Algebra

Question

Simplify:

a. $y^{\frac{3}{4}}·y^{\frac{5}{8}}$ b. ${\left(m^9\right)}^{\frac{2}{9}}$ c. $\frac{d^{{\displaystyle \frac{1}{5}}}}{d^{{\displaystyle \frac{6}{5}}}}$

Step-by-Step Solution

Answer

Part (a). The answer is $y^{\frac{11}{8}}$.

Part (b). The answer is $m$.

Part (c). The answer is $\frac{1}{d}$

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,

$$\begin{gathered} =y^{\frac{3}{4}}·y^{\frac{5}{8}} \\ =y^{\frac{3}{4}+\frac{5}{8}} \\ =y^{\frac{6+5}{8}} \\ =y^{\frac{11}{8}} \end{gathered}$$

Part (b) Step 1. Simplifying the exponent

We know that, ${\left(a^b\right)}^c=a^{b·c}$

Similarly,

$$\begin{gathered} {\left(m^9\right)}^{\frac{2}{9}} \\ {m}^{9\times \frac{2}{9}} \\ m \end{gathered}$$

Step 3 / 3 Part (c) Step 1. Simplifying the exponent

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

Similarly,

 $$\begin{gathered} \frac{d^{{\displaystyle \frac{1}{5}}}}{d^{{\displaystyle \frac{6}{5}}}} \\ {d}^{\frac{1}{5}-\frac{6}{5}} \\ {d}^{\frac{1-6}{5}} \\ {d}^{\frac{-5}{5}} \\ {d}^{-1} \\ \frac{1}{d} \end{gathered}$$