Question

Comment on the type of number: $\sqrt[5]{54}$

• A. Quadratic surd.
• B. Surd
• C. Irrational number
• D. Real number

Answer 1

Answer: (B), (C), (D)

Real number

Check for Irrational number: Let's assume $\sqrt[5]{54}$ is a rational number and its fractional form is $\frac{p}{q},$ where $p$ and $q$ are integers, $q\ne 0$ and $p$ and $q$ have no common factors other than $1.$

$\sqrt[5]{54}=\frac{p}{q}$
(raising fifth power on both sides)
$\Rightarrow (4^{\frac{1}{5}}{)}^5=\frac{p^5}{q^5}\Rightarrow 4q^5=p^5\Rightarrow 2\times 2q^5=p^5$

Here, $p$ should be an even number as $p^5$ is an even number.
$\because {\text{even}}^5=\text{even number}.$
$\Rightarrow p=2k$
(substituting this value in the original expression)

$2\times 2q^5=p^5\Rightarrow 2\times 2q^5=(2k{)}^5\Rightarrow 2\times 2q^5=32k^5\Rightarrow q^5=2\times 4k^5$
$\therefore q$ should also be an even number, which contradicts the assumption that $p$ and $q$ do not share any common factors other than $1.$

Hence, $\sqrt[5]{54}$ is an irrational number.

Check for Surd: $n^{th}$ root of a number which can not be simplified into a rational number is called surd.

Here $\sqrt[5]{54}$ can not be simplified into a rational number hence, it is a surd.

Check for Quadratic Surd: Square root of any non-perfect square is called quadratic surd.
$\sqrt[5]{54}$ is not a quadratic surd as the $5^{th}$ root is not a square root.

Also, an irrational number is a real number.