Question

Simplify the expression $1÷{(5+\frac{8}{\sqrt{2}}-\sqrt[3]{3125})}^{\frac{1}{3}}\times \sqrt{4}$ and comment whether it is a surd or not.

• A. It is a surd.
• B. It is a quadratic surd.
• C. It is not a surd.

Answer 1

The correct option is C It is not a surd.

Let's simplify the given expression using PEMDAS rule.

$\frac{1}{{(5+\frac{8}{\sqrt{2}}-\sqrt[3]{3125})}^{\frac{1}{3}}}\times \sqrt{4}=\frac{2}{{(\frac{8}{\sqrt{2}}+5-\sqrt[3]{35^3})}^{\frac{1}{3}}}=\frac{2}{{(\frac{8}{\sqrt{2}}+5-5)}^{\frac{1}{3}}}=\frac{2}{{\left(\frac{8}{\sqrt{2}}\right)}^{\frac{1}{3}}}=\frac{2\times (\sqrt{2}{)}^{\frac{1}{3}}}{{\left(8\right)}^{\frac{1}{3}}}=\frac{2\times (\sqrt{2}{)}^{\frac{1}{3}}}{2}$

$=\sqrt[6]{62}$

Now, $\sqrt[6]{62}$ can not be expressed as $\frac{p}{q},$ where $p$ and $q$ are integers, $q\ne 0$ and the ratio $\frac{p}{q}$ is in its reduced form.

$\therefore \sqrt[6]{62}$ is a surd.