PUMPA - THE SMART LEARNING APP

Helps you to prepare for any school test or exam

In the previous grade, we have learnt about exponent. Let us recall them.

We can write the number $$729$$ as $$9 \times 9 \times 9 = 9^3$$. Here, the number $$9$$ is the base and $$3$$ is the exponent. The exponent is also called as index.

Here, we have found the value of $$9^3$$. Similarly, we can find the value of $$9^{-3}$$ which is the multiplicative inverse of $$9^3$$. That is, $$9^3 \times 9^{-3} = 9^{3 - 3} = 9^0 = 1$$.

Hence, we can write $$9^{-3}$$ as $$9^{-3} = \frac{1}{9^{3}}$$
In general, we can say that $$x^{-n} = \frac{1}{x^n}$$
Consider the numbers with some powers like ${a}^{m},{b}^{n},...$

Here $$a$$ and $$b$$ the base and $$m$$ and $$n$$ are its respective exponents.

The law of exponents are as follows:

$\begin{array}{l}\left(i\right)\phantom{\rule{0.147em}{0ex}}{a}^{m}\cdot {a}^{n}={a}^{m+n}\\ \left(\mathit{ii}\right)\phantom{\rule{0.147em}{0ex}}{\left({a}^{m}\right)}^{n\phantom{\rule{0.147em}{0ex}}}=\phantom{\rule{0.147em}{0ex}}{a}^{\mathit{mn}}\\ \left(\mathit{iii}\right)\phantom{\rule{0.147em}{0ex}}\frac{{a}^{m}}{{a}^{n}}={\left(a\right)}^{m-n},\phantom{\rule{0.147em}{0ex}}\mathit{where}\phantom{\rule{0.147em}{0ex}}m\phantom{\rule{0.147em}{0ex}}>\phantom{\rule{0.147em}{0ex}}n.\\ \left(\mathit{iv}\right)\phantom{\rule{0.147em}{0ex}}{a}^{m}{b}^{m}=\phantom{\rule{0.147em}{0ex}}{\left(\mathit{ab}\right)}^{m}\end{array}$
Example:
1. Consider ${3}^{5}\cdot {3}^{-6}$,

Here the base(3) is the same but the exponents $$5$$ and $$-6$$ are different.

Comparing the property ${a}^{m}\cdot {a}^{n}={a}^{m+n}$ with the expression,

${3}^{5}\cdot {3}^{-6}\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}{3}^{5-6}={3}^{-1}=\frac{1}{3}$.

2. Let us take another expression ${\left({9}^{3}\right)}^{5}$.

Here the base (9) is the same, but the exponents $$3$$ and $$5$$ are different.

Comparing the property ${\left({a}^{m}\right)}^{n\phantom{\rule{0.147em}{0ex}}}=\phantom{\rule{0.147em}{0ex}}{a}^{\mathit{mn}}$ with the expression,

${\left({9}^{3}\right)}^{5}={9}^{3×5}={9}^{15}$.

3. Let the next expression be $\frac{{2}^{7}}{{4}^{3}}$.

Here the base values are $$2$$ and $$4 = 2^2$$, and the exponents are $$7$$ and $$3$$.

$\frac{{2}^{7}}{{4}^{3}}=\frac{{2}^{7}}{{\left({2}^{2}\right)}^{3}}$

Applying the property ${\left({a}^{m}\right)}^{n\phantom{\rule{0.147em}{0ex}}}=\phantom{\rule{0.147em}{0ex}}{a}^{\mathit{mn}}$,

$\frac{{2}^{7}}{{\left({2}^{2}\right)}^{3}}=\frac{{2}^{7}}{{2}^{6}}$

Now applying $\phantom{\rule{0.147em}{0ex}}\frac{{a}^{m}}{{a}^{n}}={\left(a\right)}^{m-n},\phantom{\rule{0.147em}{0ex}}\mathit{where}\phantom{\rule{0.147em}{0ex}}m\phantom{\rule{0.147em}{0ex}}>\phantom{\rule{0.147em}{0ex}}n.$

$\frac{{2}^{7}}{{2}^{6}}={2}^{7-6}={2}^{1}=2$