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Which of the following is an irrational number?

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\sqrt{41616}

23.232323

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

23.10100100010000 . . .

answer is D.

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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.

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