Question

√2 is a irrational prove that 5+3√2 is an irrational numbe

Answer 1

5 = rational no.
3 = rational no.
root 2 = 1.414 (irrarional no.)
therefore,by combining
we found that it is a irrational no.

Answer 2

$\huge\mathcal\pink{Rational\: number:}$

=> The numbers which can be expressed in the form of $\dfrac{p}{q}$ where " p " and " q " are integers and q ≠ 0

It is represented by " Q "

$\huge\mathcal\pink{Irrational\: number:}$

=> The numbers which can't be expressed in the form of $\dfrac{p}{q}$ where " p " and " q " are integers and q ≠ 0

It is represented by " S "

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$\huge\mathcal\pink{Solution :}$

Let us assume, to contrary that 5+ 3√2 is a rational number.

$\implies$ 5 + 3√2 = $\dfrac{p}{q}$

Where p & q are co-primes i.e H. C. F (p, q) = 1 and q ≠ 0

Now, $\dfrac{p}{q} - 5$ = 3√2

$\implies \dfrac{p - 5q}{q}$ = 3√2

$\implies \dfrac{p - 5q}{3q}$ = √2

Here, $\dfrac{p - 5q}{3q}$ is rational.

But the fact is √2 is irrational.

Rational ≠ irrational

∴ Our assumption is wrong

Hence, 5 + 3√ 2 is irrational