Question

Evaluate:

$\left(i\right)2\times {27}^{\frac{1}{3}}\times {216}^{-\frac{2}{3}}$
$\left(ii\right){64}^{-\frac{1}{3}}({64}^{\frac{1}{3}}-{64}^{-\frac{2}{3}})$ [4 MARKS]

Answer 1

Application: 2 Marks each

$\left(i\right)2\times {27}^{\frac{1}{3}}\times {216}^{-\frac{2}{3}}$

$=2\times (3^3{)}^{\frac{1}{3}}\times (6^3{)}^{-\frac{2}{3}}\left[\right(a^m{)}^n=a^{mn}]$

$=2\times \left(3^{3\times \frac{1}{3}}\right)\times \left(6^{3\times (-\frac{2}{3})}\right)$

$=2\times 3\times 6^{-2}$
$=6\times 6^{-2}$
$=6^{1-2}[a^m\times a^n=a^{m+n}]$
$=6^{-1}$
$=\frac{1}{6}$

$\left(ii\right){64}^{-\frac{1}{3}}({64}^{\frac{1}{3}}-{64}^{\frac{2}{3}})$

$=\left(4^3{)}^{-\frac{1}{3}}\left[\right(4^3{)}^{\frac{1}{3}}-\left(4^3{)}^{\frac{2}{3}}\right]\right[(a^m{)}^n=a^{mn}]$

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

$=\left(4{)}^{-1}\right(4-4^2)$

$=\frac{1}{4}(4-16)$

$=\frac{1}{4}\times -12$

$=-3$