Exponents And Powers
Quotient of Powers with the Same Exponent:
\(
\left( {\frac{x}
{y}} \right)^m = \frac{{x^m }}
{{y^m }}
\)
When you divide two numbers each raised to the same exponent, you can combine them into a single expression by dividing the numbers and raising the result to the same exponent.
L.H.S
\(
\left( {\frac{x}
{y}} \right)^m = \frac{x}
{y} \times \frac{x}
{y}..........(m\,times)
\)
\(
= \frac{{x \times x \times ........(m\,\,times)}}
{{y \times y \times ........(m\,\,times)}}
\)
\(
\left( {\frac{x}
{y}} \right)^m = \frac{{x^m }}
{{y^m }}
\)
Ex:(i)\(
\left( {\frac{2}
{3}} \right)^{13}
\)
(ii)\(
\left( {\frac{6}
{7}} \right)^{ - 1} = \frac{{6^{ - 1} }}
{{7^{ - 1} }} = 6^{ - 1} \times \frac{1}
{{7^{ - 1} }} = \frac{1}
{6} \times 7 = \frac{7}
{6}
\)
L.H.S \(
x^{ - 1} = \left( {x^{ - 1} } \right)^1 = \left( {\frac{1}
{x}} \right)^1 = \frac{1}
{x}
\) \(
\because
\) provided x > 0 and \(
x \ne 1
\)