Question

Simplify:
(i) $(27{)}^{\frac{4}{3}}+(32{)}^{0.8}+(0.8{)}^{-1}$
(ii) ${\left(\frac{64}{125}\right)}^{\frac{-2}{3}}÷\frac{1}{{\left(\frac{256}{625}\right)}^{\frac{1}{4}}}\times \frac{\sqrt{25}}{\sqrt[3]{364}}$
(iii) ${\left[5{(8^{\frac{1}{3}}+{27}^{\frac{1}{3}})}^3\right]}^{\frac{1}{4}}$

Answer 1

(i) $(27{)}^{\frac{4}{3}}+(32{)}^{0.8}+(0.8{)}^{-1}=(3^3{)}^{\frac{4}{3}}+(2^5{)}^{\frac{8}{10}}+{\left(\frac{8}{10}\right)}^{-1}$
$=3^4+2^4+\frac{10}{8}=81+16+1.25=98.25$

(ii) ${\left(\frac{64}{125}\right)}^{\frac{-2}{3}}÷\frac{1}{{\left(\frac{256}{625}\right)}^{\frac{1}{4}}}\times \frac{\sqrt{25}}{\sqrt[3]{364}}={\left(\frac{125}{64}\right)}^{\frac{2}{3}}÷{\left(\frac{625}{256}\right)}^{\frac{1}{4}}\times \frac{5}{\sqrt[3]{34^3}}$
$={\left[{\left(\frac{5}{4}\right)}^3\right]}^{\frac{2}{3}}\times {\left(\frac{256}{625}\right)}^{\frac{1}{4}}\times \frac{5}{4}={\left(\frac{5}{4}\right)}^2\times \frac{4}{5}\times \frac{5}{4}$
$=\frac{25}{16}=1\frac{9}{16}$

(iii) ${\left[5{(8^{\frac{1}{3}}+{27}^{\frac{1}{3}})}^3\right]}^{\frac{1}{4}}={\left[5{(2^{3\times \frac{1}{3}}+3^{3\times \frac{1}{3}})}^3\right]}^{\frac{1}{4}}$
$=\left[5\right(2+3{)}^3{]}^{\frac{1}{4}}=(5\times 5^3{)}^{\frac{1}{4}}=(5^4{)}^{\frac{1}{4}}=5$