Answer :

5√2 is an irrational number and subtraction of a rational number and an irrational number is an irrational number.

∴, 4 - 5√2 is irrational, where 5√2 is an irrational number while 4 is a rational number.

Alternative - Assume that 4 - 5√2 is rational

4 - 5√2 = a/b , where a and b are integers .

⇒ - 5√2 = a/b - 4

⇒ 5√2 = 4 - a/b

⇒ 5√2 = 4b/b - a/b

⇒ √2 = (4b – a) / 5b

we know that a, b, 4 and 5 are integers and they are also rational {i.e RHS is rational}

therefore √2 will be rational.

but we know that √2 is irrational.

there is a contradiction

so, 4 - 5√2 is an irrational number

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