Book Online Demo

Check Your IQ

Try Test

Courses

Offline Centres

Which of the following is an irrational number?

see full answer

Talk to JEE/NEET 2025 Toppers - Learn What Actually Works!

Real Strategies. Real People. Real Success Stories - Just 1 call away

An Intiative by Sri Chaitanya

\sqrt{41616}

23.232323

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

23.10100100010000 . . .

answer is D.

(Unlock A.I Detailed Solution for FREE)

Ready to Test Your Skills?

Check your Performance Today with our Free Mock Test used by Toppers!

Take Free Test

Detailed Solution

We need to find out an irrational number given in the options.
Considering the number

\sqrt{41616}

\sqrt{41616} = 204

Since,

\sqrt{41616}

is a perfect square,
Therefore, it is a rational number.
Considering the number

23.232323

,
It is a rational number because it is a recurring decimal.
Considering the number

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

\Rightarrow \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}}

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

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

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

Therefore, it is a rational number.
Considering the number

23.10100100010000 . . .

Since, the number 23.10100100010000 does not end.
Therefore, the number 23.10100100010000 is an irrational number.
Thus,

23.10100100010000 . . .

is an irrational number.
Hence, option 4 is correct.

Watch 3-min video & get full concept clarity

JEE

NEET

Foundation JEE

Foundation NEET

CBSE