Question

Prove that root5-root8 is irrational

Answer 1

when we minus any irrational number minus any irrational number

Answer 2

Method:

⇒ √5 - √8 = x

⇒ (√5 - √8)² = x²

⇒ 5 + 8 - 2√40 = x²

⇒ 13 - 4√10 = x²

⇒ 13 - x² = 4√10

Step-by-step explanation:

⇒ First assumed to reach our contradiction that √5 - √8 is a rational, so let √5 - √8 = x, for a rational number x. Now both LHS and RHS are rational.

⇒ Squared both sides of √5 - √8 = x. Got 13 - 4√10 = x².

⇒ Subtracted x² from both, thus got 13 - x² = 4√10.

⇒ As assumed earlier, if both sides of √5 - √8 = x are rational, then so will be 13 - x² = 4√10.

⇒ But in 13 - x² = 4√10, the LHS where a rational number x² is subtracted from another rational 13 is rational, while the RHS 4√10 is irrational.

⇒ So our assumption is contradicted!!!

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