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Question

Prove that (√3 - √2)3 is irrational


Solution

Sol :
Suppose √3 - √2 is rational .

Let √3 - √2 = r where r is a rational.

∴ (√3 - √2)^3 = r^3

∴ (2 + 3 - 2√6)(√3 - √2) = r^3
5root3-2root18-5root2+2root12=r^3


Now , LHS is many terms with irrational numbers
Because all the values inside the roots are not perfect roots

RHS = r^3 But rational number cannot be equal to an irrational.

∴our supposition is wrong.

∴ √3 - √2 is irrational .

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