Question

Simplify the following using laws of indices:
$(3^{-\frac{1}{2}}\times 2^{-\frac{1}{3}})÷(3^{-\frac{3}{4}}\times 2^{-\frac{5}{6}})$

Answer 1

The given expression $(3^{-\frac{1}{2}}\times 2^{-\frac{1}{3}})÷(3^{-\frac{3}{4}}\times 2^{-\frac{5}{6}})$ can be simplified as follows:

$(3^{-\frac{1}{2}}\times 2^{-\frac{1}{3}})÷(3^{-\frac{3}{4}}\times 2^{-\frac{5}{6}})=(3^{-\frac{1}{2}}÷3^{-\frac{3}{4}})\times (2^{-\frac{1}{3}}÷2^{-\frac{5}{6}})=3^{-\frac{1}{2}-(-\frac{3}{4})}\times 2^{-\frac{1}{3}-(-\frac{5}{6})}(\because x^m÷x^n=x^{m-n})$

$=3^{-\frac{1}{2}+\frac{3}{4}}\times 2^{-\frac{1}{3}+\frac{5}{6}}=3^{\frac{-2+3}{4}}\times 2^{\frac{-2+5}{6}}=3^{\frac{1}{4}}\times 2^{\frac{3}{6}}=3^{\frac{1}{4}}\times 2^{\frac{1}{2}}$

$=3^{\frac{1}{4}}\times 2^{\frac{2}{4}}=(3\times 2^2{)}^{\frac{1}{4}}=(3\times 4{)}^{\frac{1}{4}}={12}^{\frac{1}{4}}$

Hence $(3^{-\frac{1}{2}}\times 2^{-\frac{1}{3}})÷(3^{-\frac{3}{4}}\times 2^{-\frac{5}{6}})={12}^{\frac{1}{4}}$