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.