Question

If x=3+2√2 then,find the value of x^1/2 + x^1/2

Answer 1

Answer:

√x + 1/√x = 2√2

Solution:

• Given : x = 3 + 2√2
• To find : √x + 1/√x = ?

We have ;

=> x = 3 + 2√2

=> x = 2 + 1 + 2√2

=> x = 2 + 2√2 + 1

=> x = (√2)² + 2•√2•1 + 1²

=> x = (√2 + 1)²

=> √x = √2 + 1

Now,

If √x = √2 + 1

Thus,

=> 1/√x = 1/(√2 + 1)

=> 1/√x = (√2 - 1)/(√2 + 1)(√2 - 1)

=> 1/√x = (√2 - 1)/[(√2)² - 1²]

=> 1/√x = (√2 - 1)/(2 - 1)

=> 1/√x = (√2 - 1)/1

=> 1/√x = √2 - 1

Now,

=> √x + 1/√x = (√2 + 1) + (√2 - 1)

=> √x + 1/√x = √2 + 1 + √2 - 1

=> √x + 1/√x = 2√2

Hence,

Required answer is ; 2√2

Answer 2

$x = 3 + 2 \sqrt{2} \\ \frac{1}{x} = \frac{1}{3 - 2 \sqrt{2} } \\ \frac{1}{x} = \frac{3 - 2 \sqrt{2} }{9 - 8} \\ \frac{1}{x} = 3 - 2 \sqrt{2} \\ \\ x + \frac{1}{x} = 6 \\ x + \frac{1}{x} + 2 = 8 \\ ( { \sqrt{x} + \frac{1}{ \sqrt{x} } })^{2} = 8 \\ \sqrt{x} + \frac{1}{ \sqrt{x} } = 2 \sqrt{2}$

.