Question

Simplify
(i) $2^{\frac{2}{3}}\times 2^{\frac{1}{3}}$
(ii) $2^{\frac{2}{3}}\times 2^{\frac{1}{5}}$
(iii) $7^{\frac{5}{6}}\times 7^{\frac{2}{3}}$
(iv) $(1296{)}^{\frac{1}{4}}\times (1296{)}^{\frac{1}{2}}$

Answer 1

Since, we have $a^m\times a^n=a^{m+n}$ .......(i)

(i) Given $2^{\frac{2}{3}}\times 2^{\frac{1}{3}}=2^{\frac{2}{3}+\frac{1}{3}}$ [from equation(i)]
$=2^{\frac{3}{3}}$
$=2^1$
$\therefore 2^{\frac{2}{3}}\times 2^{\frac{1}{3}}=2$
(ii) $2^{\frac{2}{3}}\times 2^{\frac{1}{5}}=2^{\frac{2}{3}+\frac{1}{5}}$ [from equation(i)]
$=2^{\frac{10+3}{15}}$
$\therefore 2^{\frac{2}{3}}\times 2^{\frac{1}{5}}=2^{\frac{10+3}{15}}$
(iii) $7^{\frac{5}{6}}\times 7^{\frac{2}{3}}$
$7^{\frac{5}{6}}\times 7^{\frac{2}{3}}=7^{\frac{5}{6}+\frac{2}{3}}$ [from equation (i)]
$=7^{\frac{5+4}{6}}$
$=7^{\frac{9}{6}}$
$\therefore 7^{\frac{5}{6}}\times 7^{\frac{2}{3}}=7^{\frac{3}{2}}$
(iv) $(1296{)}^{\frac{1}{4}}\times (1296{)}^{\frac{1}{2}}=(1296{)}^{\frac{1}{4}+\frac{1}{2}}$
$=(1296{)}^{\frac{1+2}{4}}$ [from equation(i)]
$=(1296{)}^{\frac{3}{4}}$
$=(6^4{)}^{\frac{3}{4}}$ [Since, $(a^m{)}^n=a^{m\times n}$]
$=6^{\frac{4\times 3}{4}}$
$=6^3$
$\therefore {1296}^{\frac{1}{4}}\times {1296}^{\frac{1}{2}}=216$