5 Fractional indices

So far we have used indices that are whole numbers. We now consider fractional powers. Consider the expression ${\left(16^{\frac{1}{2}}\right)}^2$ . Using the third law of indices, ${\left(a^m\right)}^n=a^{mn}$ , we can write

${\left(16^{\frac{1}{2}}\right)}^2=16^{\frac{1}{2}\times 2}=16^1=16$

So $16^{\frac{1}{2}}$ is a number which when squared equals 16, that is 4 or $-4$ . In other words $16^{\frac{1}{2}}$ is a square root of 16. There are always two square roots of a non-zero positive number, and we write $16^{\frac{1}{2}}=\pm 4$

Similarly

${\left(8^{\frac{1}{3}}\right)}^3=8^{\frac{1}{3}\times 3}=8^1=8$

so that $8^{\frac{1}{3}}$ is a number which when cubed equals 8. Thus $8^{\frac{1}{3}}$ is the cube root of 8, that is $\sqrt[3]{8}$ , namely 2. Each number has only one cube root, and so

$8^{\frac{1}{3}}=2$

In general

More generally we have

Your calculator will be able to evaluate fractional powers, and roots of numbers. Check that you can obtain the results of the following Examples on your calculator, but be aware that calculators normally give only one root when there may be others.

Example 22

Evaluate

1. $144^{1∕2}$ ,
2. $125^{1∕3}$

Solution

1. $144^{1∕2}$ is a square root of 144, that is $\pm 12.$
2. Noting that $5^3=125$ , we see that $125^{1∕3}=\sqrt[3]{125}=5$

Example 23

Evaluate

1. $32^{1∕5}$ ,
2. $32^{2∕5}$ ,
3. $8^{2∕3}$ .

Solution

1. $32^{\frac{1}{5}}$ is the 5th root of 32, that is $\sqrt[5]{32}$ . Now $2^5=32$ and so $\sqrt[5]{32}=2$ .
2. Using the third law of indices we can write $32^{2∕5}=32^{2\times \frac{1}{5}}={\left(32^{\frac{1}{5}}\right)}^2$ . Thus

   $32^{2∕5}={\left({\left(32\right)}^{1∕5}\right)}^2=2^2=4$

3. Note that $8^{1∕3}=2$ . Then

   $8^{\frac{2}{3}}=8^{2\times \frac{1}{3}}={\left(8^{1∕3}\right)}^2=2^2=4$

   Note the following alternatives:

   $8^{2∕3}={\left(8^{1∕3}\right)}^2={\left(8^2\right)}^{1∕3}$

Example 24

Write the following as a simple power with a single index:

1. $\sqrt{x^5}$ ,
2. $\sqrt[4]{x^3}$ .

Solution

1. $\sqrt{x^5}={\left(x^5\right)}^{\frac{1}{2}}$ . Then using the third law of indices we can write this as $x^{5\times \frac{1}{2}}=x^{\frac{5}{2}}$ .
2. $\sqrt[4]{x^3}={\left(x^3\right)}^{\frac{1}{4}}$ . Using the third law we can write this as $x^{3\times \frac{1}{4}}=x^{\frac{3}{4}}$ .

Example 25

Show that $z^{-1∕2}=\frac{1}{\sqrt{z}}$ .

Solution

$z^{-1∕2}=\frac{1}{z^{1∕2}}=\frac{1}{\sqrt{z}}$

Task!

Simplify $\frac{\sqrt{z}}{z^3{z}^{-1∕2}}$

First, rewrite $\sqrt{z}$ using an index and simplify the denominator using the first law of indices:

Answer

Answer

Example 26

The generalisation of the third law of indices states that ${\left(a^m{b}^n\right)}^k=a^{mk}{b}^{nk}$ . By taking $m=1$ , $n=1$ and $k=\frac{1}{2}$ show that $\sqrt{ab}=\sqrt{a}\sqrt{b}$ .

Solution

Taking $m=1$ , $n=1$ and $k=\frac{1}{2}$ gives ${\left(ab\right)}^{1∕2}=a^{1∕2}{b}^{1∕2}$ .

Taking the case when all these roots are positive, we have $\sqrt{ab}=\sqrt{a}\sqrt{b}$ .

This result often allows answers to be written in alternative forms. For example, we may write $\sqrt{48}$ as $\sqrt{3\times 16}=\sqrt{3}\sqrt{16}=4\sqrt{3}$ .

Although this rule works for multiplication we should be aware that it does not work for addition or subtraction so that

$\sqrt{a\pm b}\ne \sqrt{a}\pm \sqrt{b}$

Exercises

1. Evaluate using a calculator
   1. $3^{1∕2}$ ,
   2. $15^{-\frac{1}{3}}$ ,
   3. $85^3$ ,
   4. $81^{1∕4}$
2. Evaluate using a calculator
   1. $15^{-5}$ ,
   2. $15^{-2∕7}$
3. Simplify
   1. $\frac{a^{11}{a}^{3∕4}}{a^{-1∕2}}$ ,
   2. $\frac{\sqrt{z}}{z^{3∕2}}$ ,
   3. $\frac{z^{-5∕2}}{\sqrt{z}}$ ,
   4. $\frac{\sqrt[3]{a}}{\sqrt[2]{a}}$ ,
   5. $\frac{\sqrt[5]{z}}{z^{1∕2}}$ .
4. Write each of the following expressions with a single index:
   1. ${\left(x^{-4}\right)}^3$ ,
   2. $x^{1∕2}{x}^{1∕4}$ ,
   3. $\frac{x^{1∕2}}{x^{1∕4}}$

Answer