Question

If X=9+4✓5, find ✓x + 1/✓x

Answer 1

Answer:

2√5

Step-by-step explanation:

Given that ;

x = 9 + 4√5

It can be written as ;

⇒ x = 4 + 5 + 4√5

⇒ x = (2)² + (√5)² + 2 * 2 * √5

⇒ x = (2 + √5)²

Also,

⇒ √x = √(2 + √5)²

⇒ √x = 2 + √5

Find the value of 1/√x ;

$\frac{1}{ \sqrt{x} } = \frac{1}{2 + \sqrt{5} } \\ \\ \frac{1}{ \sqrt{x} } = \frac{1}{2 + \sqrt{5} } \times \frac{2 - \sqrt{5} }{2 - \sqrt{5} } \\ \\ \frac{1}{ \sqrt{x} } = \frac{2 - \sqrt{5} }{(2) {}^{2} - ( \sqrt{5}) {}^{2} } \\ \\ \frac{1}{ \sqrt{x} } = \frac{2 - \sqrt{5} }{4 - 5} \\ \\ \frac{1}{ \sqrt{x} } = - (2 - \sqrt{5} ) \\ \\ \frac{1}{ \sqrt{x} } = \sqrt{5} - 2$

Find the value of √x + 1/√x,

$\sqrt{x} + \frac{1}{ \sqrt{x} } = 2 + \sqrt{5} + \sqrt{5} - 2 \\ \\\sqrt{x} + \frac{1}{ \sqrt{x} } = \sqrt{5} + \sqrt{5} \\ \\ \boxed{ \bf \therefore \: \sqrt{x} + \frac{1}{ \sqrt{x} } = 2 \sqrt{5}}$