Question

Simplify and express the result in its simplest form

(i) $\sqrt[6]{24}÷\left(\sqrt[3]{3}·\sqrt{2}\right)$

(ii) $\sqrt[3]{24}÷\left(\sqrt[4]{2}·\sqrt[3]{3}\right)$

(iii) $27\sqrt[3]{18}÷3\sqrt[3]{9}$

(iv) $\sqrt{x^3{y}^3}÷\sqrt{x^4{y}^3}$

(v) $\sqrt{m^2{n}^2}\times \sqrt[6]{m^2{n}^2}\times \sqrt[3]{m^2{n}^2}$

Answer 1

$$\begin{gathered} \left(\mathrm{i}\right)\sqrt[6]{24}÷\left(\sqrt[3]{3}\times \sqrt{2}\right) \\ =\frac{\sqrt[6]{24}}{\sqrt[3]{3}\times \sqrt{2}} \\ =\frac{\sqrt[6]{24}}{\sqrt[3\times 2]{3^2}\times \sqrt[2\times 3]{2^3}} \\ =\frac{\sqrt[6]{24}}{\sqrt[6]{9}\times \sqrt[6]{8}} \\ =\sqrt[6]{\frac{24}{9\times 8}}=\sqrt[6]{\frac{1}{3}} \end{gathered}$$

$$\begin{gathered} \left(\mathrm{ii}\right)\sqrt[3]{24}÷\left(\sqrt[4]{2}\times \sqrt[3]{3}\right) \\ =\frac{\sqrt[3]{8}\times \sqrt[3]{3}}{\sqrt[4]{2}\times \sqrt[3]{3}} \\ =\frac{\sqrt[3]{8}}{\sqrt[4]{2}} \\ =\frac{\sqrt[3\times 4]{8^4}}{\sqrt[4\times 3]{2^3}} \\ =\frac{\sqrt[12]{4096}}{\sqrt[12]{8}} \\ =\sqrt[12]{\frac{4096}{8}} \\ =\sqrt[12]{512} \end{gathered}$$



$\left(\mathrm{iii}\right)27\sqrt[3]{18}÷3\sqrt[3]{9}=\frac{27\times \sqrt[3]{18}}{3\times \sqrt[3]{9}}=9\sqrt[3]{\frac{18}{9}}=9\sqrt[3]{2}$

$\left(\mathrm{iv}\right)\sqrt[]{x^3{y}^3}÷\sqrt[3]{x^4{y}^3}=\frac{\sqrt[2\times 3]{{\left(x^3{y}^3\right)}^3}}{\sqrt[3\times 2]{{\left(x^4{y}^3\right)}^2}}=\sqrt[6]{\frac{x^9{y}^9}{x^8{y}^6}}=\sqrt[6]{x{y}^3}$

$$\begin{gathered} \left(\mathrm{v}\right)\sqrt[]{m^2{n}^2}\times \sqrt[6]{m^2{n}^2}\times \sqrt[3]{m^2{n}^2} \\ =\sqrt[2\times 3]{{\left(m^2{n}^2\right)}^3}\times \sqrt[6]{m^2{n}^2}\times \sqrt[3\times 2]{{\left(m^2{n}^2\right)}^2} \\ =\sqrt[6]{m^6{n}^6}\times \sqrt[6]{m^2{n}^2}\times \sqrt[6]{m^4{n}^4} \\ =\sqrt[6]{m^6{n}^6\times m^2{n}^2\times m^4{n}^4} \\ =\sqrt[6]{m^{12}{n}^{12}} \\ =m^2{n}^2 \end{gathered}$$