Category : 7th Class
Exponents and Powers
\[{{10}^{5}}\]is the exponential form of 1,00,000, since 1,00,000 =10\[\times \]10\[\times \]10\[\times \]10\[\times \]10.
In \[{{10}^{5}},\,10\] is the base and 5 is the exponent or index or power.
\[a\times a={{a}^{2}}\](read as 'a squared' or 'a raised to the power 2')
\[a\times a\times a={{a}^{3}}\](read as 'a cubed' or 'a raised to the power 3')
\[a\times a\times a\times a={{a}^{4}}\] (read as 'a raised to the power 4' or \[{{4}^{th}}\] power of a)
…………………………………………………….
\[a\times a\times a\,....\] (n factors) \[={{a}^{n}}\](read as 'a raised to the power n' or \[{{\operatorname{n}}^{th}}\]power of a)
e.g.,\[{{\left( -5 \right)}^{4}}=\left( -5 \right)\times \left( -5 \right)\times \left( -5 \right)\times \left( -5 \right)=+\,625\]
e.g., \[{{\left( -3 \right)}^{5}}=\left( -3 \right)\times \left( -3 \right)\times \left( -3 \right)x\left( -3 \right)x\left( -3 \right)=\left( -\,243 \right)\]
Note: (a) \[{{\left( -1 \right)}^{oddnumber}}=-1\]
(b)\[{{\left( -1 \right)}^{oddnumber}}=+1\]
For any non-zero integers 'a' and V and whole numbers 'm' and 'n',
(i) \[a\times a\times a\times ......\times a\](m factors)\[={{a}^{m}}\]
(ii) \[{{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}\]
(iii) \[\frac{{{a}^{m}}}{{{a}^{n}}}={{a}^{m+n}},\operatorname{if}\,\,m>n\]
\[=1,\text{ }if\text{ }m=n\]
\[=\frac{1}{{{a}^{n-m}}}\,\operatorname{if}\,m<n\]
(iv) \[{{\left( {{a}^{m}} \right)}^{n}}{{a}^{mn}}\]
(v) \[{{\left( ab \right)}^{m}}={{a}^{m}}{{b}^{m}}\]
(vi) \[{{\left( \frac{a}{b} \right)}^{m}}=\frac{{{a}^{m}}}{{{b}^{m}}}\]
(vii)\[{{a}^{o}}=1\]
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