Question

if x=3+2√2 then find the value of √x-1/√x​

Answer 1

Answer:

2-√2/3

Step-by-step explanation:

mark brainlist

Answer 2

$\green\bigstar$Answer:

$\sf \boxed{ \sqrt{x} \: - \: \dfrac{1}{ \sqrt{x} \: } = \: 2}$

$\pink\bigstar$Given:

• x = 3 + 2√2

$\red\bigstar$To find:

• $\sf \sqrt{x} \: - \: \dfrac{1}{ \sqrt{x} }$

$\red\bigstar$ Solution:

Given that x = 3 + 2√2

So, $\sqrt{x}$ = $\sqrt{3\: + \: 2 \sqrt{2}}$

$\sf \: \implies \: \sqrt{x} \: = \: \sqrt{2 \: + \:( 2 \: \times \: 1 \: \times \sqrt{2} ) \: + \: 1}$

$\sf \implies \sqrt{x} = \sqrt{( { \sqrt{2} })^{2} \: + \: (2 \times \sqrt{2} \times 1)+ \: {(1)}^{2} }$

(By using (a + b)² = a² + 2ab + b²)

$\sf \: \implies \: \sqrt{x} \: = \sqrt{ {( \sqrt{2} \: + \: 1 )}^{2}}$

$\sf \implies \: \sqrt{x} \: = \: ( \sqrt{2} \: + \: 1)$

$\sf \boxed{ \sf\therefore \: \sqrt{x} \: = \: \sqrt{2} \: + \: 1}$

Now let's find the value of $\dfrac{1}{ \sqrt{x} }$

$\sf\dfrac{1}{ \sqrt{x} } \: = \: \dfrac{1}{ \sqrt{2} \: + \: 1 }$

By rationalising the denominator,

$\sf \implies\dfrac{1}{ \sqrt{x} } \: = \: \dfrac{1 \: \times ( \sqrt{2} - 1) }{ (\sqrt{2} + 1)( \sqrt{2} - 1)}$

( By using (a + b) (a - b) = a² - b² )

$\sf\implies\dfrac{ 1}{ \sqrt{x} } \: = \: \dfrac{ \sqrt{2} - 1}{ ({ \sqrt{2} )}^{2} \: - \: {(1)}^{2} }$

( $\because$ √a × √a = a )

$\sf\implies\dfrac{1}{ \sqrt{x} } \: = \: \dfrac{ \sqrt{2} - 1}{ 2 - 1}$

$\sf\implies\dfrac{1}{ \sqrt{x} } \: = \dfrac{ \sqrt{2} \: - \: 1 } {1}$

$\sf \boxed{ \sf\implies\dfrac{1}{ \sqrt{x} } \: = { \sqrt{2} \: - \: 1 } }$

$\sf Now, \sqrt{x} - \dfrac{1}{\sqrt{x}}$ :

$\sf \: Now \: by \: substituting \: the \: values \: of \: \sqrt{x} \: and \: \dfrac{1}{ \sqrt{x} }, \: we \: get$

$= \sf\: \sqrt{2} \: + \: 1 \: - \: ( \sqrt{2} \: - \: 1 \: )$

$= \sf\: \sqrt{2} \: + \: 1 \: - \: \sqrt{2} \: + \: 1$

$= \: \sf \cancel{ \sqrt{2} } \: + \: 1 \: \cancel{- \sqrt{2}} \: + \: 1$

$= \sf \: 1 \: + \: 1$

$= \sf \boxed{ 2 }$

$\sf \boxed{\therefore \sqrt{x} \: - \: \dfrac{1}{ \sqrt{x} \: } = \: 2}$

$\pink\bigstar$ Concepts Used:

• Factorisation of polynomial
• Cancellation of root and power
• Rationalisation of Denominator
• (a + b)(a - b) = a² - b²
• (a + b)² = a² + 2ab + b²
• √a × √a = a
• Negative sign × Negative sign = Positive sign
• Cancellation of numbers with opposite signs

$\blue\bigstar$Extra Information:

• (a + b)² = a² + 2ab + b²
• (a – b)² = a² – 2ab + b²

• a² – b² = (a + b)(a – b)

• (x + a)(x + b) = x² + (a + b) x + ab

• (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca

• (a + b)³ = a³ + b³ + 3ab (a + b)

• (a – b)³ = a³ – b³ – 3ab (a – b)

• a³ + b³ + c³ – 3abc = (a + b + c)(a² + b² + c² – ab – bc – ca)