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Question

State which of the following are pure surds and mixed surds: (i) $\sqrt[5]{7}$ (ii) $\sqrt{21}$ (iii) $\sqrt[3]{34}$ (iv) $\frac{3}{4}\sqrt{8}$ (v) $\sqrt{1350}$ (vi) $5\sqrt[3]{135}$ (vii) $\sqrt[10]{3}$ (viii) $\sqrt{9.3}$ (ix) $\sqrt{0.4}$ (x) $\sqrt{\frac{49}{10}}$

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Solution

A surd of the form $k\sqrt[n]{a}$ (a) is a mixed surd if $k\in Qandk\ne ±1$. (b) is a pure surd if $k\in Qandk=±1$. (i) $5\sqrt{7}\mathrm{is}\mathrm{a}\mathrm{mixed}\mathrm{surd}\mathrm{since}\mathrm{k}=5\left(i.e.k\ne ±1\right)$. (ii) $\sqrt{21}\mathrm{is}\mathrm{a}\mathrm{pure}\mathrm{surd}\mathrm{since}\mathrm{k}=1\left(i.e.k=±1\right)$. (iii)$\sqrt[3]{34}\mathrm{is}\mathrm{a}\mathrm{pure}\mathrm{surd}\mathrm{since}\mathrm{k}=1\left(i.e.k=±1\right)$. (iv) $\frac{3}{4}\sqrt{8}\mathrm{is}\mathrm{a}\mathrm{mixed}\mathrm{surd}\mathrm{since}\mathrm{k}=\frac{3}{4}\left(i.e.k\ne ±1\right)$. (v) $\sqrt{1350}\mathrm{is}\mathrm{a}\mathrm{pure}\mathrm{surd}\mathrm{since}\mathrm{k}=1\left(i.e.k=±1\right)$. (vi) $5\sqrt[3]{135}\mathrm{is}\mathrm{a}\mathrm{mixed}\mathrm{surd}\mathrm{since}\mathrm{k}=5\left(i.e.k\ne ±1\right)$. (vii) $\sqrt[10]{3}\mathrm{is}\mathrm{a}\mathrm{pure}\mathrm{surd}\mathrm{since}\mathrm{k}=1\left(i.e.k=±1\right)$. (viii) $\sqrt{9.3}\mathrm{is}\mathrm{a}\mathrm{pure}\mathrm{surd}\mathrm{since}\mathrm{k}=1\left(i.e.k=±1\right)$. (ix) $\sqrt{0.4}\mathrm{is}\mathrm{a}\mathrm{pure}\mathrm{surd}\mathrm{since}\mathrm{k}=1\left(i.e.k=±1\right)$. (x) $\sqrt{\frac{49}{10}}\mathrm{is}\mathrm{a}\mathrm{pure}\mathrm{surd}\mathrm{since}\mathrm{k}=1\left(i.e.k=±1\right)$.

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