8th Class Mathematics Exponents and Power Exponents and Powers

 Exponents and Powers

• Exponential equation: An equation which has an unknown quantity as an exponent is called an exponential equation.

e.g.,      (i)\[{{5}^{x}}=625\]

(ii) \[{{3}^{x-5}}=1\]

Note: If ax = ay, than x = y.

• Standard form of numbers: A number written in the form \[\left( m\times {{10}^{n}} \right)\]is said to be in standard form if 'm' is a decimal number between 1 and 9 and 'n' is either a positive or a negative integer.

Very large numbers and very small numbers are expressed in standard form.

• Laws of exponents (Integers): For any two non-zero integers 'a' and 'b', and any integers 'm' and 'n', the following laws hold good.

(i) \[{{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}\]     

            (ii)\[\frac{{{a}^{m}}}{{{a}^{n}}}={{a}^{m+n}}\left( m>n \right)\]

            (iii) \[\frac{{{a}^{m}}}{{{a}^{n}}}=\frac{1}{{{a}^{m+n}}}\left( m<n \right)\]

            (iv)\[\frac{{{a}^{m}}}{{{a}^{n}}}={{a}^{0}}\left( m=n \right)\]

            (v)\[{{\left( {{a}^{m}} \right)}^{n}}={{a}^{mn}}\]

            (vi)\[{{a}^{m}}\times {{b}^{m}}={{\left( ab \right)}^{m}}\]

            (vii) \[\frac{{{a}^{m}}}{{{b}^{m}}}={{\left( \frac{a}{b} \right)}^{m}}\]

            (viii) \[{{a}^{0}}=1\]

• Positive integral exponent of a rational number: For any rational number\[\frac{a}{b}\]and a positive integer\['n',{{\left( \frac{a}{b} \right)}^{n}}=\frac{{{a}^{n}}}{{{b}^{n}}}\].

• Negative integral exponent of a rational number: For any rational number. \[\frac{a}{b}\] and a

positive integer \['n',{{\left( \frac{a}{b} \right)}^{-n}}={{\left( \frac{b}{a} \right)}^{n}}\].

• Zero exponent of a rational number: For any rational number\[\frac{a}{b}\]and '0',\[{{\left( \frac{a}{b} \right)}^{0}}=1\].

• Special case of\[{{\mathbf{a}}^{\mathbf{n}}}\]:

\[{{a}^{n}}=1\]Only if n = 0 for any 'a' except a = 1 or a = -1.

For a = 1,

\[{{1}^{1}}={{1}^{2}}={{1}^{3}}={{1}^{-2}}=....=1\]or\[{{\left( 1 \right)}^{n}}=1\]for infinitely many \['n',\]

• For a = - 1,

\[{{\left( -1 \right)}^{0}}={{\left( -1 \right)}^{2}}={{\left( -1 \right)}^{4}}={{\left( -1 \right)}^{-2}}=....=1\,or{{\left( -1 \right)}^{P}}=1\]for any even integer\['P',\] and \[{{(-1)}^{q}}=(-1)\] for any odd integer 'q'.

• Laws of exponents (Rational numbers): Let\[\frac{a}{b}\,and\,\frac{c}{d}\]be any two rational numbers, and 'm' and 'n' be any integers. Then,

\[{{\left( \frac{a}{b} \right)}^{m}}\times {{\left( \frac{a}{b} \right)}^{n}}={{\left( \frac{a}{b} \right)}^{m+n}}\]

(ii) \[{{\left( \frac{a}{b} \right)}^{m}}+{{\left( \frac{a}{b} \right)}^{n}}={{\left( \frac{a}{b} \right)}^{m-n}}\]

(iii) \[{{\left\{ {{\left( \frac{a}{b} \right)}^{m}} \right\}}^{n}}={{\left( \frac{a}{b} \right)}^{mn}}\]

(iv) \[{{\left( \frac{a}{b}\times \frac{c}{d} \right)}^{n}}={{\left( \frac{a}{b} \right)}^{n}}\times {{\left( \frac{c}{d} \right)}^{n}}and\,{{\left\{ \frac{a/b}{c/d} \right\}}^{n}}=\frac{{{\left( a/b \right)}^{n}}}{{{\left( c/d \right)}^{n}}}\]

(v) \[{{\left( \frac{a}{b} \right)}^{-n}}={{\left( \frac{b}{a} \right)}^{0}},\]When \['n'\]is a positive integer.

(vi) \[{{\left( \frac{a}{b} \right)}^{0}}=1\]