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) $\sqrt{\frac{49}{10}}$

Answer 1

A surd of the form $k\sqrt[n]{a}$

(a) is a mixed surd if $k\in Qandk\ne \pm 1$.
(b) is a pure surd if $k\in Qandk=\pm 1$.


(i) $5\sqrt{7}\mathrm{is}\mathrm{a}\mathrm{mixed}\mathrm{surd}\mathrm{since}\mathrm{k}=5(i.e.k\ne \pm 1)$.

(ii) $\sqrt{21}\mathrm{is}\mathrm{a}\mathrm{pure}\mathrm{surd}\mathrm{since}\mathrm{k}=1(i.e.k=\pm 1)$.

(iii)$\sqrt[3]{34}\mathrm{is}\mathrm{a}\mathrm{pure}\mathrm{surd}\mathrm{since}\mathrm{k}=1(i.e.k=\pm 1)$.

(iv) $\frac{3}{4}\sqrt{8}\mathrm{is}\mathrm{a}\mathrm{mixed}\mathrm{surd}\mathrm{since}\mathrm{k}=\frac{3}{4}(i.e.k\ne \pm 1)$.

(v) $\sqrt{1350}\mathrm{is}\mathrm{a}\mathrm{pure}\mathrm{surd}\mathrm{since}\mathrm{k}=1(i.e.k=\pm 1)$.

(vi) $5\sqrt[3]{135}\mathrm{is}\mathrm{a}\mathrm{mixed}\mathrm{surd}\mathrm{since}\mathrm{k}=5(i.e.k\ne \pm 1)$.

(vii) $\sqrt[10]{3}\mathrm{is}\mathrm{a}\mathrm{pure}\mathrm{surd}\mathrm{since}\mathrm{k}=1(i.e.k=\pm 1)$.

(viii) $\sqrt{9.3}\mathrm{is}\mathrm{a}\mathrm{pure}\mathrm{surd}\mathrm{since}\mathrm{k}=1(i.e.k=\pm 1)$.

(ix) $\sqrt{0.4}\mathrm{is}\mathrm{a}\mathrm{pure}\mathrm{surd}\mathrm{since}\mathrm{k}=1(i.e.k=\pm 1)$.

(x) $\sqrt{\frac{49}{10}}\mathrm{is}\mathrm{a}\mathrm{pure}\mathrm{surd}\mathrm{since}\mathrm{k}=1(i.e.k=\pm 1)$.