Question

(√2-2)², Classify the given number as rational or irrational.

Answer 1

Given number = ( √2 - 2 )²

This is of the form ( a - b )²

We know that ( a - b )² = a² - 2ab + b²

=> ( √2 - 2 )² = ( √2 )² - 2 ( √2 ) ( 2 ) + 2²

=> ( √2 - 2 )² = 2 - 4√2 + 4

=> ( √2 - 2 )² = 6 - 4√2

Since 6 - 4√2 is an irrational number ( As anything added with irrational number is an irrational number ), ( √2 - 2 )² is also an irrational number.

Answer 2

Given ( √2 - 2 )²

= (√2)²-2×√2×2 + 2²

[ By algebraic identity ,

( x - y )² = x² - 2xy + y² ]

= 2 - 4√2 + 4

= 6 - 4√2

[ Irrational number ]

Therefore ,

( √2 - 2 )² is an irrational number.

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