1.3.6: Exponents and Square Roots of Fractions

Learning Objectives

Simplify fractions raised to a power.
Simplify square roots of fractions.

Key words

Base: The number being raised to a power in an exponential expression
Exponent: The power in an exponential expression
Principal square root: The positive square root
Negative square root: The negative square root

Exponents

Exponents are used as a concise way to write multiple multiplication. For example, $2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2=2^{7}$ . The exponent $7$ tells us how many times we have to multiply the base $2$ by itself.

This same notation is used to raise a fraction to a power.

Examples

Identify the base and the exponent in each term:

1. $\left ( \frac{2}{7}\right )^{6}$ Base = $\frac{2}{7}$ and exponent = $6$

2. $\left ( \frac{-5}{9}\right )^{4}$ Base = $\frac{-5}{9}$ and exponent = $4$

3. $\left ( -\frac{1}{25}\right )^{3}$ Base = $-\frac{1}{25}$ and exponent = $3$

Try It

Identify the base and the exponent in each term:

1. $\left ( \frac{4}{3}\right )^{5}$ Base = $\frac{4}{3}$ and exponent = $5$

2. $\left ( \frac{-1}{8}\right )^{7}$ Base = $\frac{-1}{8}$ and exponent = $7$

3. $\left ( -\frac{11}{5}\right )^{3}$ Base = $-\frac{11}{5}$ and exponent = $3$

Examples

1. $\left(\frac{2}{3}\right )^{2}$ = $\frac{2}{3}\cdot\frac{2}{3}=\frac{4}{9}$

2. $\left (\frac{-1}{2}\right )^{3}$ = $\frac{-1}{2}\cdot\frac{-1}{2}\cdot\frac{-1}{2}=\frac{-1}{8}$

3. $\left (-\frac{3}{4}\right )^{4}$ = $-\frac{3}{4}\cdot -\frac{3}{4}\cdot -\frac{3}{4}\cdot -\frac{3}{4}=\frac{81}{64}$

4. $\left(\frac{1}{2}\right )^{5}=\frac{1}{2}\cdot\frac{1}{2}\cdot\frac{1}{2}\cdot\frac{1}{2}\cdot\frac{1}{2}=\frac{1}{32}$

5. $\left(\frac{-2}{5}\right )^{3}=\frac{-2}{5}\cdot\frac{-2}{5}\cdot\frac{-2}{5}=\frac{-8}{125}$

6. $\left(-\frac{2}{7}\right )^{2}=-\frac{2}{7}\cdot -\frac{2}{7}=\frac{4}{49}$

Try It

Simplify by writing the exponential term as multiple multiplication:

1. $\left(\frac{4}{5}\right )^{2}$

2. $\left (\frac{-2}{5}\right )^{3}$

3. $\left (-\frac{2}{3}\right )^{4}$

Show Answer

1. $\frac{4}{5}\cdot\frac{4}{5}=\frac{16}{25}$

2. $\frac{-2}{5}\cdot\frac{-2}{5}\cdot\frac{-2}{5}=\frac{-8}{125}$

3. $-\frac{2}{3} \cdot -\frac{2}{3}\cdot -\frac{2}{3}\cdot -\frac{2}{3}=\frac{16}{81}$

Square Roots

The principal square root of a positive integer is the positive number that multiplies by itself to give the integer. As we saw in the section on integers, a positive integer has two square roots. The principal square root is positive and uses the radical sign for notation. For example, $\sqrt{4}=2$ . The integer $4$ is a perfect square since its square root is a whole number. $4$ also has a negative square root notated with a negative sign in front of the radical: $-\sqrt{4}=-2$ .

To find the square root of a fraction, we look for the fraction that squares to give the original fraction.

For example, to find the square root of $\frac{4}{9}$ we are looking for a fraction that when squared gives us $\frac{4}{9}$ . Since $2^2 = 4$ and $3^{2} =9$ , $\frac{2^{2}}{3^{2}}=\frac{4}{9}$ . Therefore, $\sqrt{\frac{4}{9}}=\frac{2}{3}$ .

Notice that $\sqrt{4}=2$ and $\sqrt{9}=3$ . This means that $\sqrt{\frac{4}{9}}=\frac{\sqrt{4}}{\sqrt{9}}=\frac{2}{3}$ .

SQUARE ROOT RULE FOR FRACTIONS

$\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$ for any whole numbers $a, b;\; b \ne 0$ .

Examples

Simplify:

1. $\sqrt{\frac{25}{16}}=\frac{\sqrt{25}}{\sqrt{16}}=\frac{5}{4}$

2. $-\sqrt{\frac{81}{4}}=-\frac{\sqrt{81}}{\sqrt{4}}=-\frac{9}{2}$

3. $\sqrt{\frac{121}{144}}=\frac{\sqrt{121}}{\sqrt{144}}=\frac{11}{12}$

4. $\sqrt{\frac{-9}{25}}=\frac{\sqrt{-9}}{\sqrt{25}}$ is undefined in the set of real numbers

Try It

Simplify:

1. $\sqrt{\frac{36}{49}}$

2. $\sqrt{\frac{100}{25}}$

3. $-\sqrt{\frac{16}{121}}$

4. $\sqrt{\frac{-16}{81}}$

Show Answer

1. $\frac{6}{7}$

2. $2$

3. $-\frac{4}{11}$

4. Undefined

Chapter 1: Real Numbers and Sets