Powers of Fractions

$powers of fractions$

All these are used in our day to day life. We use these problems to calculate certain things, like the distance of Mars from the planet Earth, calculating in light-years.

An exponent is a number that is raised to a base. But what do we mean by an Exponent?

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Let us just refresh our memory on rules of exponents and importantly focus on the “fraction rule” before we go further.

What are the rules for exponents?

Exponent is a short version of expressing how many times a number is supposed to be multiplied by itself.

How do you solve fractions with exponents?

Just like integers with exponent powers, we solve fractions with exponent powers.

Now let’s take a look at this example:

$\left ( \frac{2}{{3}} \right )^3$ = $\frac{2}{3}$ x $\frac{2}{3}$ x $\frac{2}{3}$ = $\frac{8}{27}$

Now how is this done?

The numerators get multiplied by itself and the denominators get multiplied by itself.

$\left ( \frac{4}{{7}} \right )^2$ = $\frac{4\ast4}{7\ast7}$ = $\frac{16}{49}$

Examples:

$\left ( \frac{1}{{3}} \right )^3$ = $\frac{1}{3}$ x $\frac{1}{3}$ x $\frac{1}{3}$
$\left ( \frac{2}{{5}} \right )^4$ = $\frac{2}{5}$ x $\frac{2}{5}$ x $\frac{2}{5}$ x $\frac{2}{5}$ = $\frac{16}{625}$

How do you solve fraction exponents?

$\frac{1}{2}$ as an exponent means square root.

$\frac{1}{3}$ as an exponent means cube root.

$\frac{1}{4}$ as an exponent means fourth root and so on.

Example:

$(9)^{\frac{1}{2}}$ = $\sqr{9}$ =3

$(343)^{\frac{1}{3}}$ = $\sqrt[3]{343}$ =7

$(16)^{\frac{1}{4}}$ = $\sqrt[4]{16}$ = 2

$(8)^{\frac{4}{3}}$ = $\left ( 2 ^{3} \right )^{\frac{4}{3}}$ = $2^4$ =16

Check Point

Solve:

$\left ( \frac{5}{9} \right )^2$
$\left ( \frac{3}{5} \right )^3$
$(81)^{\frac{1}{4}}$
$(256)^{\frac{5}{8}}$
$\left ( \frac{243}{32} \right )^{\frac{2}{5}}$

Answer key

$\frac{25}{81}$
$\frac{27}{125}$
3
32
$\frac{9}{4}$

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Power Of Fractions

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