Question

Simplify:

(i) $\left(\sqrt{3}+\frac{1}{\sqrt{3}}\right)\left(\sqrt{3}-\frac{1}{\sqrt{3}}\right)$

(ii) $\left(\sqrt[4]{6}-\sqrt[4]{5}\right)\left(\sqrt[4]{6}+\sqrt[4]{5}\right)\left(\sqrt[]{6}+\sqrt[]{5}\right)$

(iii) $\left(\sqrt{5}+\sqrt{2}\right)\left(\sqrt{5}+3\sqrt{2}\right)$

(iv) $2\sqrt{x^3}\times x\sqrt[3]{a}$

(v) $\sqrt[3]{2}\times \sqrt{5}$

(vi) $\sqrt{2}.\sqrt[3]{3}.\sqrt[4]{4}$

(vii) $\sqrt[6]{12}÷\left(\sqrt{3}\sqrt[3]{2}\right)$

(viii) $\sqrt[4]{28}÷\sqrt[3]{7}$

(ix) $6\sqrt[3]{25}÷\sqrt[2]{5}$

(x) $\sqrt[6]{256}÷\sqrt[3]{2}$

(xi) $\sqrt{x^6{y}^3}÷\sqrt[3]{x^{2}y}$

Answer 1

$$\begin{gathered} \left(\mathrm{i}\right)\left(\sqrt{3}+\frac{1}{\sqrt{3}}\right)\left(\sqrt{3}-\frac{1}{\sqrt{3}}\right)={\left(\sqrt{3}\right)}^2-{\left(\frac{1}{\sqrt{3}}\right)}^2 \\ =3-\frac{1}{3} \\ =\frac{8}{3} \end{gathered}$$

$$\begin{gathered} \left(\mathrm{ii}\right)\left(\sqrt[4]{6}-\sqrt[4]{5}\right)\left(\sqrt[4]{6}-\sqrt[4]{5}\right)\left(\sqrt[]{6}+\sqrt[]{5}\right)=\left[{\left(\sqrt[4]{6}\right)}^2-{\left(\sqrt[4]{5}\right)}^2\right]\left(\sqrt{6}+\sqrt{5}\right) \\ =\left(\sqrt{6}-\sqrt{5}\right)\left(\sqrt{6}+\sqrt{5}\right) \\ ={\left(\sqrt{6}\right)}^2-{\left(\sqrt{5}\right)}^2 \\ =6-5 \\ =1 \end{gathered}$$

$$\begin{gathered} \left(\mathrm{iii}\right)\left(\sqrt{5}+\sqrt{2}\right)\left(\sqrt{5}+3\sqrt{2}\right)=\sqrt{5}\left(\sqrt{5}+3\sqrt{2}\right)+\sqrt{2}\left(\sqrt{5}+3\sqrt{2}\right) \\ =5+3\sqrt{10}+\sqrt{10}+6 \\ =11+4\sqrt{10} \end{gathered}$$

$$\begin{gathered} \left(\mathrm{iv}\right)2\sqrt[]{x^3}\times x\sqrt[3]{a}=2\sqrt[2\times 3]{{\left(x^3\right)}^2}\times x\sqrt[3\times 2]{a^2} \\ =2\sqrt[6]{x^6}\times x\sqrt[6]{a^2} \\ =2x\sqrt[6]{a^2{x}^6} \\ =2x\sqrt[6]{{\left(a{x}^3\right)}^2} \\ =2x\sqrt[3]{a{x}^3} \\ =2\sqrt[3]{x^3}\sqrt[3]{a{x}^3} \\ =2\sqrt[3]{a{x}^6} \end{gathered}$$

$$\begin{gathered} \left(\mathrm{v}\right)\sqrt[3]{2}\times \sqrt{5}=\sqrt[3\times 2]{2^2}\times \sqrt[2\times 3]{5^3} \\ =\sqrt[6]{4}\times \sqrt[6]{125} \\ =\sqrt[6]{500} \end{gathered}$$

$$\begin{gathered} \left(\mathrm{vi}\right)\sqrt{2}\times \sqrt[3]{3}\times \sqrt[4]{4}=\sqrt[2\times 6]{2^6}\times \sqrt[3\times 4]{3^4}\times \sqrt[4\times 3]{4^3} \\ =\sqrt[12]{64}\times \sqrt[12]{81}\times \sqrt[12]{64} \\ =\sqrt[12]{331776} \end{gathered}$$

$$\begin{gathered} \left(\mathrm{vii}\right)\sqrt[6]{12}÷\left(\sqrt{3}\times \sqrt[3]{2}\right)=\frac{\sqrt[6]{12}}{\sqrt[2\times 3]{3^3}\times \sqrt[3\times 2]{2^2}} \\ =\frac{\sqrt[6]{12}}{\sqrt[6]{27}\times \sqrt[6]{4}} \\ =\sqrt[6]{\frac{12}{27\times 4}} \\ =\sqrt[6]{\frac{1}{9}} \\ =\sqrt[6]{\frac{1}{3^2}}=\sqrt[3]{\frac{1}{3}}=\frac{1}{\sqrt[3]{3}} \end{gathered}$$

$$\begin{gathered} \left(\mathrm{viii}\right)4\sqrt{28}÷3\sqrt{7}=\frac{4\sqrt{28}}{3\sqrt{7}} \\ =\frac{4}{3}\times \sqrt{\frac{28}{7}} \\ =\frac{4}{3}\times \sqrt{4} \\ =\frac{4}{3}\times 2 \\ =\frac{8}{3} \end{gathered}$$

$$\begin{gathered} \left(\mathrm{ix}\right)6\sqrt[3]{25}÷2\sqrt{5}=\frac{6\sqrt[3]{25}}{2\sqrt{5}} \\ =\frac{6\sqrt[3\times 2]{{25}^2}}{2\sqrt[2\times 3]{5^3}} \\ =\frac{6\times \sqrt[6]{625}}{2\times \sqrt[6]{125}} \\ =3\sqrt[6]{\frac{625}{125}} \\ =3\sqrt[6]{5} \end{gathered}$$

$$\begin{gathered} \left(\mathrm{x}\right)\sqrt[6]{256}÷\sqrt[3]{2}=\frac{\sqrt[6]{256}}{\sqrt[3]{2}} \\ =\frac{\sqrt[6]{256}}{\sqrt[3\times 2]{2^2}} \\ =\frac{\sqrt[6]{256}}{\sqrt[6]{4}} \\ =\sqrt[6]{\frac{256}{4}} \\ =\sqrt[6]{64} \\ =\sqrt[6]{2^6} \\ =2 \end{gathered}$$

$$\begin{gathered} \left(\mathrm{xi}\right)\sqrt[]{x^6{y}^3}÷\sqrt[3]{x^{2}y}=\frac{\sqrt{x^6{y}^3}}{\sqrt[3]{x^{2}y}} \\ =\frac{\sqrt[2\times 3]{{\left(x^6{y}^3\right)}^3}}{\sqrt[3\times 2]{{\left(x^{2}y\right)}^2}} \\ =\frac{\sqrt[6]{x^{18}{y}^9}}{\sqrt[6]{x^4{y}^2}} \\ =\sqrt[6]{\frac{x^{18}{y}^9}{x^4{y}^2}} \\ =\sqrt[6]{x^{14}{y}^7} \end{gathered}$$