Category :
8th Class
EXPONENTS & POWERS
FUNDAMENTALS
Exponent:-
- The exponent of a number says how many times it should be used in a multiplication.
Example:- \[3\,in\,{{2}^{3}}\Rightarrow {{2}^{3}}=2\times 2\times 2;\]
\[5\,in\,{{2}^{5}}\Rightarrow {{2}^{5}}=2\times 2\times 2\times 2\times 2\]
- Exponent is also called power or index or indices
- \[{{x}^{y}}\]can be read as yth power of x (or) x raised to the power y.
- \[2\times 2\times 2\times 2\times 2={{2}^{5}}\],
Here \[2\times 2\times 2\times 2\times 2\] is called the product form (or) expanded form and \[{{2}^{5}}\] is called the exponential form.
- Product Rule:- \[{{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}\] (Where \[a~\ne 0\]be any rational number and m, n be rational numbers)
Example:- \[{{2}^{5}}\times {{2}^{6}}={{\left( 2 \right)}^{5+6}}={{2}^{11}}\]
- Quotient Rule:- \[\frac{{{a}^{m}}}{{{a}^{n}}}~={{a}^{m-n}}\] (Where \[a~\ne 0\]be any rational number )
- Example:- \[{{5}^{6}}\div {{5}^{3}}={{5}^{6-3}}={{5}^{3}}\]
- \[{{({{a}^{m}})}^{n}}={{a}^{m\times n}}\] (Where m, n are rational numbers, \[a~\ne 0\])
Example:- \[{{\left( {{3}^{5}} \right)}^{6}}={{3}^{5\times 6}}={{3}^{30}}\]
- \[{{\left( \frac{a}{b} \right)}^{m}}=\frac{{{a}^{m}}}{{{b}^{m}}},\] (Where \[a~\ne 0\], \[b\ne 0\])
Note:-
- The first power of a number is the number itself. i.e., \[{{\mathbf{a}}^{\mathbf{1}}}=\mathbf{a}\]
- The second power is called square. E.g., square of ‘5’ is \[{{5}^{2}}\]
- The third power is called cube. E.g., cube of y is \[{{y}^{3}}\]
Number in Standard form:-
- A number written as \[(x\times {{10}^{y}})\] is said to be in standard form if x is a decimal number such that \[1\le x\le 10\] and y is either a positive or a negative integer.
- Expressing large numbers in standard form
E.g., Express 246784 in standard form
Solution:- \[246784=2.46784\times 100000\]
\[=\left( 2.4678\times {{10}^{5}} \right)\]
Expressing very small numbers is standard form
E.g., Express 0.00003 in standard form.
- \[{{\left( \frac{2}{3} \right)}^{2}}=\frac{2}{3}\times \frac{2}{3}=\frac{{{2}^{2}}}{{{3}^{2}}}.\]
- \[{{a}^{m}}\times {{b}^{m}}={{(ab)}^{m}}\] (\[ab\ne 0\] and m is positive integer)
e.g., \[{{2}^{3}}\times {{3}^{3}}={{\left( 2\times 3 \right)}^{m}}={{6}^{3}}.\]
- \[{{(a)}^{0}}=1\,\,(where\,a\ne 0)\]
e.g., \[{{\left( 3 \right)}^{0}}=1\]
- \[{{(a)}^{-n}}={{\left( \frac{1}{a} \right)}^{n}}(where\,a\ne 0)\]
e.g., \[{{(2)}^{-3}}=\frac{1}{{{2}^{3}}}=\frac{1}{8}.\]
\[\Rightarrow m=n\] if
\[a\ne 0,-1,1\]
e.g., \[{{2}^{3}}={{2}^{x}}\]
\[\therefore x=3\]
Rule for One
- 1 raised to any integral power gives E.g., \[{{(1)}^{1000}}=1\]
- (-1) raised to any odd natural number gives '-1’ E.g., \[{{(1)}^{3789}}=-1\]
- (-1) raised to any even natural number gives '+1' or simply 1. E.g., \[{{(-1)}^{4628}}=~1\]
