Question

Choose the correct statement(s).

• A. Any number that can be expressed as $\sqrt[n]{na}$ form, is a surd.
• B. Any number that can be expressed as $\frac{p}{q}$ form, where $p$ and $q$ are integers and $q\ne 0,$ is irrational.
• C. Euler's number ${}^{\prime }{e}^{\prime }$ is an irrational number.

Answer 1

The correct option is C Euler's number ${}^{\prime }{e}^{\prime }$ is an irrational number.

Let's go through the given statements one by one.

$\underset{–}{1^{st}\text{statement}}$

When a number in the form $\sqrt[n]{na}$ cannot be simplified further into a rational number. it is a surd.
For example, $\sqrt[3]{32}$ is surd or an irrational number.
Whereas, $\sqrt[3]{3125}$ is not a surd or an irrational number as,
$\sqrt[3]{35\times 5\times 5}=5=\frac{5}{1}$

$\therefore 1^{st}\text{statement}$ is not always true.

$\underset{–}{2^{nd}\text{statement}}$ is wrong as any number that can be expressed as $\frac{p}{q}$ form, where $p$ and $q$ are integers and $q\ne 0,$ is rational.

$\underset{–}{3^{rd}\text{statement}}$ is correct.

Value of ${}^{\prime }{e}^{\prime }=\mathrm{2.718...}$
It is non-terminating and non-repeating. Hence, it is an irrational number.