We looked at exponents in Class 8 as well. Before starting this topics, related to Class 9, it is a good idea to quickly revisit what we learned in 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) $a^n = a \times a \times a \times ... \times a \ (n \ factors)$

Examples: $2^3 = 2 \times 2 \times 2 = 8$ $(\frac{3}{2})^4 = \frac{3}{2} \times \frac{3}{2} \times \frac{3}{2} \times \frac{3}{2} = \frac{91}{16}$ $\\$

b) $a^0 = 1$

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

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

Examples: $7^{-3} =$ $\frac{1}{7^3}$ $($ $\frac{3}{2}$ $)^{-2} =$ $\frac{1}{(\frac{3}{2})^2}$ $=$ $\frac{1}{\frac{3}{2} \times \frac{3}{2}}$ $=$ $\frac{4}{9}$ $\big($ $\frac{1}{5}$ $\big)^{-2} =$ $\frac{1}{(\frac{1}{5})^2} = \frac{1}{\frac{1}{25}}$ $= 25$ $\\$

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

Examples: $\frac{5^8}{5^4}$ $= 5^{8-4} = 5^4 = 625$ $\frac{2^4}{2^2}$ $= 2^{4-2} = 2^2 = 4$ $\\$

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

Examples: $(3^2)^5 = 3^{2 \times 5} = 3^{10}$ $\bigg\{ \Big\{ \frac{2}{3}\Big\}^4\bigg\} ^3 = \Big\{ \frac{2}{3}\Big\} ^{4 \times 3} = \Big\{ \frac{2}{3} \Big\}^{12}$ $\\$

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

Examples: $6^4 = (2 \times 3)^4 = 2^4 \times 3^4$ $(\frac{2}{3} \times \frac{3}{4})^3 = (\frac{2}{3})^3 \times (\frac{3}{4})^3 \ = \frac{1}{8}$ $\\$

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

Examples: $(\frac{2}{3})^3$ $=$ $\frac{2^3}{3^3}$ $(\frac{-4}{5})^5$ $=$ $\frac{(-4)^5}{5^5}$ $\\$

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

Examples: $2^{\frac{1}{2}}$ $= \sqrt{2}$ $3^{\frac{1}{3}}$ $= \sqrt{3}$ $\\$

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

Examples: $2^2 \times 2^3 = 2^{2+3} = 2^5$ $3^3 \times 3^{\frac{1}{2}} = 3^{3+\frac{1}{2}} = 3^{\frac{7}{2}}$ $\\$

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

Examples: $5^{\frac{2}{3}}$ $= (\sqrt{5})^2$ $7^{\frac{3}{5}}$ $= (\sqrt{7})^3$ $\\$

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