Exponents And Powers
Product of Powers with the Same Exponent:
\(
\left( {xy} \right)^m = x^m \times y^m
\)
When you multiply two numbers each raised to the same exponent, you can combine them into a single expression by multiplying the numbers and raising the result to the same exponent.
L.H.S \(
= (xy)^m
\)
\(
= xy \times xy \times .......m\,\,times
\)
\(
= x \times y \times x \times y.......m\,\,times
\)
\(
= x \times x \times .......(m\,\,times)\,\, \times \,\,(y \times y.......m\,times)
\)
\(
\left( {xy} \right)^m = x^m \times y^m
\)
Ex:- 1.\(
\left( {\sqrt 2 .\sqrt 3 } \right)^5 = \left( {\sqrt 2 } \right)^5 .\left( {\sqrt 3 } \right)^5
\)
2.\(
\left( {\frac{3}
{2}.\frac{7}
{5}} \right)^{11} = \left( {\frac{3}
{2}} \right)^{11} .\left( {\frac{7}
{5}} \right)^{11}
\)