Question

Prove that 5 - 2√5 is an irrational number.

Answer 1

hey dear

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here is your answer

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Let √2 + √5 be rational number

A rational number can be written in the form of p / q integer unequal to 0

√2 + √5 = p / q

squaring on both the sides we get

( √2 + √5 ) ^2 = ( p / q ) ^2

( √2 )^2 + ( √5 ) ^2 + 2 ( √2 ) (√5) = p^2 / q ^2

2 + 5 +2 √10 = p ^2. / q ^2

7 + 2√10 = p^2. / q ^2.

2√10 = p^2 / q ^2. - 7

√10. = ( p^2 - 7q^2 ) / 2

p, q are integers then ( p^2 - 7q^2 ) / 2. are rational number

so √10 is also a rational number

but it contradicts our facts that √10 is a rational number

so, our supposition is false

hence √2 +√5 is irrational number

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Answer 2

Proved below.

Given: The expression 5 - 2√5

To prove: 5 - 2√5 is irrational

Solution:

Let us assume 5 - 2√5 is rational

⇒ 5 - 2√5 = $\frac{p}{q}$, where p and q are integers and q ≠ 0

⇒ 2√5 = 5 - $\frac{p}{q}$

⇒ √5 = $\frac{5q - p}{2q}$

⇒ √5 is rational

But we know that √5 is an irrational number.

⇒ Contradiction due to incorrect assumption

⇒ 5 - 2√5 is not irrational

⇒ 5 - 2√5 is rational

Hence proved by the contradiction method.

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