Question

Show that 2root3 + 5 is an irrational number.​

Answer 1

$\underline{\underline{\sf{\maltese\:\:Question}}}$

Show that 2√3 + 5 is an irrational number.

$\underline{\underline{\sf{\maltese\:\:To\: prove}}}$

• 2√3 + 5 is an irrational number.

$\underline{\underline{\sf{\maltese\:\:Proof}}}$

Let 2√3 + 5 be a rational number.

A rational number can be expressed as $\frac{a}{b}$, where b is not equal to 0 and a and b are integers.

Then,

$\longrightarrow \: 2 \sqrt{3} + 5 = \frac{a}{b}$

$\longrightarrow \: 2 \sqrt{3} = \frac{a}{b} - 5$

$\longrightarrow \: \sqrt{3} = \frac{1}{2} (\frac{a}{b} - 5)$

Here,

R.H.S = $\frac{1}{2} (\frac{a}{b} - 5)$ and it is rational.

while, L.H.S, √3 is irrational which is not possible.

Hence, our assumption that 2√3 + 5 is a rational number is wrong. Hence, 2√3 + 5 is an irrational number.

HENCE PROVED!!

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