Q 153E - Intermediate Algebra

Question

In the following exercises, simplify:

Part (a): $y^{\frac{1}{2}}.y^{\frac{3}{4}}$

Part (b): ${\left(x^{12}\right)}^{\frac{2}{3}}$

Part (c): $\frac{m^{{\displaystyle \frac{5}{8}}}}{m^{{\displaystyle \frac{13}{8}}}}$

Step-by-Step Solution

Answer

Part (a): On simplifying, we get $y^{\frac{5}{4}}$.

Part (b): On simplifying, we get $x^8$.

Part (c): On simplifying, we get $\frac{1}{m}$.

Part (a) Step 1. Given information.

Consider the given question,

$y^{\frac{1}{2}}.y^{\frac{3}{4}}$

Using the product property,

$=y^{\frac{1}{2}+\frac{3}{4}}$

Add the fractions,

$$\begin{gathered} =y^{\frac{2+3}{4}} \\ =y^{\frac{5}{4}} \end{gathered}$$

Part (b) Step 1. Given information.

Consider the given question,

${\left(x^{12}\right)}^{\frac{2}{3}}$

Using the power property,

$$\begin{gathered} =x^{12\times \frac{2}{3}} \\ =x^{4\times 2} \\ =x^8 \end{gathered}$$

Part (c) Step 1. Given information.

Consider the given question,

$\frac{m^{{\displaystyle \frac{5}{8}}}}{m^{{\displaystyle \frac{13}{8}}}}$

Using the quotient property,

$$\begin{gathered} =m^{\frac{5}{8}-\frac{13}{8}} \\ =m^{-\frac{8}{8}} \\ =m^{-1} \\ =\frac{1}{m} \end{gathered}$$