Category : 8th Class
Exponents and Powers
e.g., (i)\[{{5}^{x}}=625\]
(ii) \[{{3}^{x-5}}=1\]
Note: If ax = ay, than x = y.
Very large numbers and very small numbers are expressed in standard form.
(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 integer \['n',{{\left( \frac{a}{b} \right)}^{-n}}={{\left( \frac{b}{a} \right)}^{n}}\].
\[{{a}^{n}}=1\]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 \['n',\]
\[{{\left( -1 \right)}^{0}}={{\left( -1 \right)}^{2}}={{\left( -1 \right)}^{4}}={{\left( -1 \right)}^{-2}}=....=1\,or{{\left( -1 \right)}^{P}}=1\]for any even integer\['P',\] and \[{{(-1)}^{q}}=(-1)\] for any odd integer 'q'.
\[{{\left( \frac{a}{b} \right)}^{m}}\times {{\left( \frac{a}{b} \right)}^{n}}={{\left( \frac{a}{b} \right)}^{m+n}}\]
(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) \[{{\left( \frac{a}{b} \right)}^{0}}=1\]
You need to login to perform this action.
You will be redirected in
3 sec