Which of the following is an irrational number?

Foundation

Mathematics Foundation

Properties of Real Number

Irrational Number

Rational Numbers

Question

Which of the following is an irrational number?

A

23.232323

B

23.10100100010000.....

C

$\frac{(1 + \sqrt{3})^{3} - (1 - \sqrt{3})^{3}}{\sqrt{3}}$

D

$\sqrt{41616}$

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Op-(A)

\sqrt{41616} = 204

\sqrt{41616}

is a perfect square, so it is a rational number.

Op-(B)

23.232323....

It is recurring decimal, so it is a rational number.

Op-(C)

\frac{(1 + \sqrt{3})^{3} - (1 - \sqrt{3})^{3}}{\sqrt{3}} = \frac{2 \sqrt{3} (1 + 2 \sqrt{3} + 3 - 2 + 1 - 2 \sqrt{3} + 3)}{\sqrt{3}} = \frac{2 \sqrt{3} (6)}{\sqrt{3}} = \frac{12 \sqrt{3}}{\sqrt{3}} = 12

.

Only, the number 23.10100100010000....... is not terminating.

So, D is the irrational number.