Question
Express each one of the following with rational denominator: $\frac{30}{5\sqrt3-3\sqrt5}$
✓
Answer
$\frac{30}{5\sqrt3-3\sqrt5}$
Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factor
$5\sqrt3+3\sqrt5$
$=\frac{30\times(5\sqrt3+3\sqrt5)}{(5\sqrt3-3\sqrt5)(5\sqrt3+3\sqrt5)}$
As we know
$\because(\text{a}+\text{b})(\text{a}-\text{b})=\text{a}^2-\text{b}^2$
$=\frac{30\times(5\sqrt3+3\sqrt5)}{75-45}$
$=\frac{30\times(5\sqrt3+3\sqrt5)}{30}$
$=5\sqrt3+3\sqrt5$
Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factor
$5\sqrt3+3\sqrt5$
$=\frac{30\times(5\sqrt3+3\sqrt5)}{(5\sqrt3-3\sqrt5)(5\sqrt3+3\sqrt5)}$
As we know
$\because(\text{a}+\text{b})(\text{a}-\text{b})=\text{a}^2-\text{b}^2$
$=\frac{30\times(5\sqrt3+3\sqrt5)}{75-45}$
$=\frac{30\times(5\sqrt3+3\sqrt5)}{30}$
$=5\sqrt3+3\sqrt5$
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