Question

State which of the following are surds. Justify.

(i) $\sqrt[3]{51}$ (ii) $\sqrt[4]{16}$ (iii) $\sqrt[5]{81}$ (iv) $\sqrt{256}$ (v) $\sqrt[3]{64}$ (vi) $\sqrt{\frac{22}{7}}$

Answer 1

(i) Since, $\sqrt[3]{51}={\left(3\times 17\right)}^{\frac{1}{3}}$

So, $\sqrt[3]{51}$ is a surd.

(ii) Since, $\sqrt[4]{16}=\sqrt[4]{2^4}=2$

So, $\sqrt[4]{16}$ is not a surd.

(iii) Since, $\sqrt[5]{81}=\sqrt[5]{3^4}={\left(3^4\right)}^{\frac{1}{5}}=3^{\frac{4}{5}}$

So, $\sqrt[5]{81}$ is a surd.

(iv) Since, $\sqrt{256}=\sqrt{{16}^2}=16$

So, $\sqrt{256}$ is not a surd.

(v) Since, $\sqrt[3]{64}=\sqrt[3]{4^3}=4$

So, $\sqrt[3]{64}$ is not a surd.

(vi) Since, $\sqrt{\frac{22}{7}}={\left(\frac{22}{7}\right)}^{\frac{1}{2}}$

So, $\sqrt{\frac{22}{7}}$ is a surd.