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}

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b)   a^0 = 1

Examples:

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

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

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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

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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

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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}

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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}

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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}

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h)   a^{\frac{1}{n}} = \sqrt[n]{a}

Examples:

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

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

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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}}

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

Examples:

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

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

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

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