Question

Simplify:

(i) $\sqrt[3]{3}+\sqrt[10]{3}$

(ii) $\sqrt[7]{5}-\sqrt[4]{5}+\sqrt{125}$

(iii) $\sqrt[4]{8}+\sqrt{32}-\sqrt[3]{2}$

(iv) $\sqrt{50}-\sqrt{98}+\sqrt{162}$

(v) $\sqrt[7]{48}-\sqrt{72}-\sqrt{27}+\sqrt[3]{2}$

(vi) $\sqrt[5]{3}+\sqrt[2]{27}+4\sqrt{\frac{1}{3}}$

(vii) $\frac{1}{2}\sqrt{243}-\sqrt{\frac{27}{4}}$

(viii) $\sqrt{\frac{8}{9}}-3\sqrt{\frac{1}{2}}+5\sqrt{\frac{9}{8}}$

(ix) $\sqrt[4]{2}-2\sqrt{8}+\frac{3}{\sqrt{2}}$

(x) $8\sqrt[3]{16}+\sqrt[3]{54}+\sqrt[3]{250}$

Answer 1

$$\begin{gathered} \left(\mathrm{i}\right)3\sqrt{3}+10\sqrt{3} \\ =\left(3+10\right)\sqrt{3} \\ =13\sqrt{3} \end{gathered}$$

$$\begin{gathered} \left(\mathrm{ii}\right)7\sqrt{5}-4\sqrt{5}+125 \\ =7\sqrt{5}-4\sqrt{5}+\sqrt{25}\times \sqrt{5} \\ =7\sqrt{5}-4\sqrt{5}+5\sqrt{5} \\ =\left(7-4+5\right)\sqrt{5} \\ =8\sqrt{5} \end{gathered}$$

$$\begin{gathered} \left(\mathrm{iii}\right)4\sqrt{8}+\sqrt{32}-3\sqrt{2} \\ =4\left(\sqrt{4}\times \sqrt{2}\right)+\sqrt{16}\times \sqrt{2}-3\sqrt{2} \\ =8\sqrt{2}+4\sqrt{2}-3\sqrt{2} \\ =\left(8+4-3\right)\sqrt{2} \\ =9\sqrt{2} \end{gathered}$$

$$\begin{gathered} \left(\mathrm{iv}\right)\sqrt{50}-\sqrt{98}+\sqrt{162} \\ =\sqrt{25}\times \sqrt{2}-\sqrt{49}\times \sqrt{2}+\sqrt{81}\times \sqrt{2} \\ =5\sqrt{2}-7\sqrt{2}+9\sqrt{2} \\ =\left(5-7+9\right)\sqrt{2} \\ =7\sqrt{2} \end{gathered}$$

$$\begin{gathered} \left(\mathrm{v}\right)7\sqrt{48}-\sqrt{72}-\sqrt{27}+3\sqrt{2} \\ =7\left(\sqrt{16}\times \sqrt{3}\right)-\sqrt{36}\times \sqrt{2}-\sqrt{9}\times \sqrt{3}+3\sqrt{2} \\ =28\sqrt{3}-6\sqrt{2}-3\sqrt{3}+3\sqrt{2} \\ =\left(28-3\right)\sqrt{3}-\left(6-3\right)\sqrt{2} \\ =25\sqrt{3}-3\sqrt{2} \end{gathered}$$


$$\begin{gathered} \left(\mathrm{vi}\right)5\sqrt{3}+2\sqrt{27}+4\sqrt{\frac{1}{3}} \\ =5\sqrt{3}+2\left(\sqrt{9}\times \sqrt{3}\right)+\times 4\times \sqrt{\frac{1}{3}}\times \frac{\sqrt{3}}{\sqrt{3}} \\ =5\sqrt{3}+6\sqrt{3}+\frac{4}{3}\sqrt{3} \\ =\left(5+6+\frac{4}{3}\right)\sqrt{3} \\ =\frac{37}{3}\sqrt{3} \end{gathered}$$

$$\begin{gathered} \left(\mathrm{vii}\right)\frac{1}{2}\sqrt{243}-\sqrt{\frac{27}{4}} \\ =\frac{1}{2}\times \sqrt{81}\times \sqrt{3}-\frac{\sqrt{9}\times \sqrt{3}}{\sqrt{4}} \\ =\frac{9}{2}\sqrt{3}-\frac{3}{2}\sqrt{3} \\ =\left(\frac{9-3}{2}\right)\sqrt{3} \\ =3\sqrt{3} \end{gathered}$$

$$\begin{gathered} \left(\mathrm{viiii}\right)\sqrt{\frac{8}{9}}-3\sqrt{\frac{1}{2}}+5\sqrt{\frac{9}{8}} \\ =\frac{\sqrt{4}\times \sqrt{2}}{\sqrt{9}}-3\times \frac{1\times \sqrt{2}}{\sqrt{2}\times \sqrt{2}}+5\times \frac{\sqrt{9\times 2}}{\sqrt{8\times 2}} \\ =\frac{2}{3}\sqrt{2}-\frac{3}{2}\sqrt{2}+\frac{15}{4}\times \sqrt{2} \\ =\left(\frac{2}{3}-\frac{3}{2}+\frac{15}{4}\right)\sqrt{2} \\ =\frac{35}{12}\sqrt{2} \end{gathered}$$


$$\begin{gathered} \left(\mathrm{ix}\right)4\sqrt{2}-2\sqrt{8}+\frac{3}{\sqrt{2}} \\ =4\sqrt{2}-2\left(\sqrt{4}\times \sqrt{2}\right)+\frac{3}{\sqrt{2}}\times \frac{\sqrt{2}}{\sqrt{2}} \\ =4\sqrt{2}-4\sqrt{2}+\frac{3}{2}\sqrt{2} \\ =\left(4-4+\frac{3}{2}\right)\sqrt{2} \\ =\frac{3}{2}\sqrt{2} \end{gathered}$$

$$\begin{gathered} \left(\mathrm{x}\right)8\sqrt[3]{16}+\sqrt[3]{54}+\sqrt[3]{250} \\ =8\times \sqrt[3]{8}\times \sqrt[3]{2}+\sqrt[3]{27}\times \sqrt[3]{2}+\sqrt[3]{125}\times \sqrt[3]{2} \\ =16\sqrt[3]{2}+3\sqrt[3]{2}+5\sqrt[3]{2} \\ =\left(16+3+5\right)\sqrt[3]{2} \\ =24\sqrt[3]{2} \end{gathered}$$