Question

Evaluate
(i) ${\left(1^3+2^3+3^3\right)}^{\frac{1}{2}}$

(ii) ${\left[5{\left(8^{\frac{1}{3}}+{27}^{\frac{1}{3}}\right)}^3\right]}^{\frac{1}{4}}$

(iii) $\frac{2^0+7^0}{5^0}$

(iv) ${\left[{\left(16\right)}^{\frac{1}{2}}\right]}^{\frac{1}{2}}$

Answer 1

(i) ${\left(1^3+2^3+3^3\right)}^{\frac{1}{2}}$
$$\begin{gathered} {\left(1^3+2^3+3^3\right)}^{\frac{1}{2}}={\left(1+8+27\right)}^{\frac{1}{2}} \\ ={\left(36\right)}^{\frac{1}{2}} \\ ={\left[{\left(6\right)}^2\right]}^{\frac{1}{2}} \\ =6 \end{gathered}$$

(ii) ${\left[5{\left(8^{\frac{1}{3}}+{27}^{\frac{1}{3}}\right)}^3\right]}^{\frac{1}{4}}$
$$\begin{gathered} {\left[5{\left(8^{\frac{1}{3}}+{27}^{\frac{1}{3}}\right)}^3\right]}^{\frac{1}{4}}={\left\{5{\left[{\left[{\left(2\right)}^3\right]}^{\frac{1}{3}}+{\left[{\left(3\right)}^3\right]}^{\frac{1}{3}}\right]}^3\right\}}^{\frac{1}{4}} \\ ={\left[5{\left(2+3\right)}^3\right]}^{\frac{1}{4}} \\ ={\left[{\left(5\right)}^4\right]}^{\frac{1}{4}} \\ =5 \end{gathered}$$

(iii) $\frac{2^0+7^0}{5^0}$
$$\begin{gathered} \frac{2^0+7^0}{5^0}=\frac{1+1}{1} \\ =2 \end{gathered}$$

(iv) ${\left[{\left(16\right)}^{\frac{1}{2}}\right]}^{\frac{1}{2}}$
$$\begin{gathered} {\left[{\left(16\right)}^{\frac{1}{2}}\right]}^{\frac{1}{2}}={\left\{{\left[{\left(4\right)}^2\right]}^{\frac{1}{2}}\right\}}^{\frac{1}{2}} \\ ={\left[{\left(4\right)}^1\right]}^{\frac{1}{2}} \\ ={\left[{\left(2\right)}^2\right]}^{\frac{1}{2}} \\ =2 \end{gathered}$$