Law of Exponents
For any two rational numbers a and b and for any integer's m and n we have:
- \[{{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}\]
- \[\frac{{{a}^{m}}}{{{a}^{n}}}={{a}^{m-n}}\]
- \[{{\left( {{a}^{m}} \right)}^{n}}={{a}^{mn}}={{\left( {{a}^{n}} \right)}^{m}}\]
- \[{{\left( ab \right)}^{n}}={{a}^{n}}{{b}^{n}}\]
- \[{{\left( \frac{a}{b} \right)}^{n}}=\frac{{{a}^{n}}}{{{b}^{n}}}\]
- If \[x\] is a rational number \[\left( x>0 \right)\] and a, b are rational exponents so that, a > b then \[{{x}^{a}}\div {{x}^{b}}={{x}^{a-b}}.\]
- If x is a rational number \[\left( x>0 \right)\] and a, b are rational exponents so that \[a<b\] then \[{{x}^{a}}\div {{x}^{b}}=\frac{1}{{{x}^{b-a}}}.\]
- If x is a rational number \[(x>0)\] and a, b and c are rational exponents then\[{{\left\{ {{\left( {{x}^{a}} \right)}^{b}} \right\}}^{c}}={{x}^{abc}}\]
- If \[x\] and y are rational numbers so that \[x>0,\text{ }y>0,\]and a is a rational exponent then \[{{x}^{a}}\times {{y}^{a}}={{\left( x\times y \right)}^{a}}.\]
