If x = 3 + 2√2 , then find the value of √x-1/√x .
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Given :x = 3 + 2√2
To find : value of √x - 1/√x
Solution :
We have , x = 3 + 2√2
x = 2 + 1 + 2√2
x = (√2)² + 1² + 2 × √2 × 1
x = (√2 + 1)²
√x = √(√2 + 1)²
√x = √2 + 1 ………(1)
1/√x = 1/√2 + 1
On rationalising the denominator :
1/√x = (1 × √2 - 1) /(√2 + 1)(√2 - 1)
1/√x = √2 - 1/(√2)² - 1²
[By using an identity, (a +b) (a - b) = a² - b²]
1/√x = √2 - 1/ (2 - 1)
1/√x = √2 - 1/ 1
1/√x = √2 - 1………(2)
Now ,
√x - 1/√x = √2 + 1 - (√2 - 1)
√x - 1/√x = √2 + 1 - √2 + 1
√x - 1/√x = 1 + 1 = 2
√x - 1/√x = 2
Hence, the value of √x - 1/√x is 2.
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