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.