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

Irrational number is the number which can not be expressed as the quotient of 2 numbers. The irrational numbers are non-terminating and non-recruiting numbers.
The numbers are non-terminating and non-recruiting numbers.
For example,

\sqrt{2}, \dots

etc.
Here,

a = \sqrt{41616}

a = \sqrt{204^{2}}

a = 204

So, it’s a rational number.
Second number is,

b = 23.232323

….. (1)
Multiply by 100,

100 b = 100 \times 23.232323

(2)
Subtract equation (1) from (2)

\begin{matrix} 99 b = 2323.2323 \dots \dots - 23.232323 . . \\ 99 b = 2300 \\ b = \frac{2300}{99} \end{matrix}

Thus, it is also an irrational number.
For the third options

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

= \frac{1 + 3 \sqrt{3} + 3 \sqrt{3} + 9 - (1 - 3 \sqrt{3} - 3 \sqrt{3} + 9)}{\sqrt{3}} = \frac{12 \sqrt{3}}{\sqrt{3}}

= 12

Which is a rational number.
Here, the number 23.10100100010000 is non-terminating and non-recruiting numbers. Thus, the irrational number is 23.10100100010000.
Hence, the correct option is 4.

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