We looked at exponents in Class 8 as well. Before starting this topic, related to Class 9, it is a good idea to quickly revisit what we learned in the previous class.

Class 8: Exponents – Lecture Notes

Class 8: Exponents – Exercise 18

Now let’s look at 9th Grade Indices (Exponents). These are the basic laws that hold true for exponents:

a) $\displaystyle a^n = a \times a \times a \times ... \times a \ (n \ factors)$

Examples:

$\displaystyle 2^3 = 2 \times 2 \times 2 = 8$

$\displaystyle (\frac{3}{2})^4 = \frac{3}{2} \times \frac{3}{2} \times \frac{3}{2} \times \frac{3}{2} = \frac{91}{16}$

$\displaystyle \\$

b) $\displaystyle a^0 = 1$

Examples:

$\displaystyle 3^0 = 1, \ \ 7^0 = 1$

$\displaystyle ( \frac{4}{3} )^0 = 1$, $\displaystyle ( - \frac{3}{7} )^0 = 1$,

$\displaystyle \\$

c) $\displaystyle a^{-n} = \frac{1}{a^n}$

Examples:

$\displaystyle 7^{-3} = \frac{1}{7^3}$

$\displaystyle ( \frac{3}{2} )^{-2} = \frac{1}{(\frac{3}{2})^2} = \frac{1}{\frac{3}{2} \times \frac{3}{2}} = \frac{4}{9}$

$\displaystyle \big( \frac{1}{5} \big)^{-2} = \frac{1}{(\frac{1}{5})^2} = \frac{1}{\frac{1}{25}} = 25$

$\displaystyle \\$

d) $\displaystyle \frac{a^m}{a^n} = a^{m-n}$

Examples:

$\displaystyle \frac{5^8}{5^4} = 5^{8-4} = 5^4 = 625$

$\displaystyle \frac{2^4}{2^2} = 2^{4-2} = 2^2 = 4$

$\displaystyle \\$

e) $\displaystyle (a^m)^n = a^{mn} = (a^n)^m$

Examples:

$\displaystyle (3^2)^5 = 3^{2 \times 5} = 3^{10}$

$\displaystyle \bigg\{ \Big\{ \frac{2}{3}\Big\}^4\bigg\} ^3 = \Big\{ \frac{2}{3}\Big\} ^{4 \times 3} = \Big\{ \frac{2}{3} \Big\}^{12}$

$\displaystyle \\$

f) $\displaystyle (ab)^n = a^n b^n$

Examples:

$\displaystyle 6^4 = (2 \times 3)^4 = 2^4 \times 3^4$

$\displaystyle (\frac{2}{3} \times \frac{3}{4})^3 = (\frac{2}{3})^3 \times (\frac{3}{4})^3 \ = \frac{1}{8}$

$\displaystyle \\$

g) $\displaystyle (\frac{a}{b})^n = \frac{a^n}{b^n}$ , $\displaystyle b \neq 0$

Examples:

$\displaystyle (\frac{2}{3})^3 = \frac{2^3}{3^3}$

$\displaystyle (\frac{-4}{5})^5 = \frac{(-4)^5}{5^5}$

$\displaystyle \\$

h) $\displaystyle a^{\frac{1}{n}} = \sqrt[n]{a}$

Examples:

$\displaystyle 2^{\frac{1}{2}} = \sqrt[2]{2}$

$\displaystyle 3^{\frac{1}{3}} = \sqrt[3]{3}$

$\displaystyle \\$

i) $\displaystyle a^m \times a^n = a^{m+n}$

Examples:

$\displaystyle 2^2 \times 2^3 = 2^{2+3} = 2^5$

$\displaystyle 3^3 \times 3^{\frac{1}{2}} = 3^{3+\frac{1}{2}} = 3^{\frac{7}{2}}$

$\displaystyle \\$

j) $\displaystyle a^{\frac{m}{n}} = (\sqrt[n]{a})^m$

Examples:

$\displaystyle 5^{\frac{2}{3}} = (\sqrt[3]{5})^2$

$\displaystyle 7^{\frac{3}{5}} = (\sqrt[5]{7})^3$

$\displaystyle \\$

Basically, these are the laws of exponents that you need to remember and apply.