Question

Express $(5^{\frac{1}{2}}+9^{\frac{1}{8}})÷(5^{\frac{1}{2}}-9^{\frac{1}{8}})$ as an equivalent fraction with a rational denominator.

Answer 1

To rationalise the denominator, which is equal to $5^{\frac{1}{2}}-3^{\frac{1}{4}}$, put $5^{\frac{1}{2}}=x$ and $3^{\frac{1}{4}}=y$; then since $x^4-y^4=(x-y)(x^3+x^{2}y+x{y}^2+y^4)$

The required factor is $5^{\frac{3}{2}}+5^{\frac{2}{2}}.3^{\frac{1}{4}}+5^{\frac{1}{2}}.3^{\frac{2}{4}}+3^{\frac{3}{4}}$;

and the rational denominator is $5^{\frac{4}{2}}-3^{\frac{4}{4}}=5^2-3=22$

$\therefore$ the expression $=\frac{(5^{\frac{1}{2}}+3^{\frac{1}{4}})(5^{\frac{3}{2}}+5^{\frac{2}{2}}.3^{\frac{1}{4}}+5^{\frac{1}{2}}.3^{\frac{2}{4}}+3^{\frac{3}{4}})}{22}$

$=\frac{5^{\frac{4}{2}}+2.5^{\frac{3}{2}}.3^{\frac{1}{4}}+2.5^{\frac{2}{2}}{3}^{\frac{2}{4}}+2.5^{\frac{1}{2}}.3^{\frac{2}{4}}+3^{\frac{4}{4}}}{22}$

$=\frac{14+5^{\frac{3}{2}}.3^{\frac{1}{4}}+5.3^{\frac{1}{2}}+5^{\frac{1}{2}}.3^{\frac{2}{4}}}{11}$