Question

Simplify the given expression and comment of the result.

$\frac{\sqrt{3}+\sqrt[3]{327}\times \frac{1}{\sqrt{3}}}{2}$

• A. It is a quadratic surd.
• B. It is an rational number.
• C. It is an irrational number.
• D. It is a real number.

Answer 1

Answer: (A), (C), (D)

It is a real number.

Let's simplify the expression using PEMDAS rule.

$\frac{\sqrt{3}+\sqrt[3]{327}\times \frac{1}{\sqrt{3}}}{2}=\frac{\sqrt{3}+(3^3{)}^{\frac{1}{3}}\times \frac{1}{\sqrt{3}}}{2}$
$=\frac{\sqrt{3}+3\times \frac{1}{\sqrt{3}}}{2}$

(as, $3=\sqrt{3}\times \sqrt{3})$

$=\frac{\sqrt{3}+\sqrt{3}\times \sqrt{3}\times \frac{1}{\sqrt{3}}}{2}$
(canceling the common terms)
$=\frac{\sqrt{3}+\sqrt{3}}{2}=\sqrt{3}$

$\sqrt{3}$ is a quadratic surd.
A quadratic surd is an irrational number, which is, in turn, a real number.