Question

34. Given that √2 is irrational, prove that (5+3√2) is an irrational number.
​

Answer 1

Answer

Given that

2

is irrational.

We know that the theorem "The product of any irrational number with a rational number is irrational".

So, since we know 3 is a rational number, 3

2

is irrational.

(3=

1

3

;1



=0)

Now we know that 3

2

is irrational.

Theorem: The sum of a rational number with an irrational number is irrational.

So, since we know 5 is a rational number, 5+3

2

is irrational.

(5=

1

5

;1



=0)

Thus we proved that 5+3

2

is an irrational number

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Answer 2

Answer:

$\huge \bf \blue{Question}$

Given that √2 is irrational,prove that (5+3√2) is an irrational number.

$\huge \bf \blue{Answer}$

Let(5+3√2) be a rational number.

5+3√2=p/q

(Where q≠0 and p and q are co-prime numbers)

3√2=p/q-5

√2=(p-5q)/3q

p and q are integers and g≠0.

(p-5q)/3q is rational number.

2 is a rational number but √2 is irrational number!

This contradiction has arisen because our assumption is wrong.So we could conclude that (5+3√2) is an irrational number.