Question

rationalising the denominator of 3+root2/3root2=a+broot2 they find the value of a and b​

Answer 1

$\blue\bigstar$ Correct Question:

Find the value of a :

$\sf\dfrac{3 \: + \: \sqrt{2} }{3 \: - \: \sqrt{2} } \: = \: a \: + \: b \sqrt{2}$

$\blue\bigstar$Answer:

$\sf \boxed{\: a \: = \: \dfrac{7}{11} \sf \: and \: \: b \: = \: \dfrac{( - 7)}{12} }$

$\pink\bigstar$ Given:

• $\sf\dfrac{3 \: + \: \sqrt{2} }{3 \: - \: \sqrt{2} } \: = \: a \: + \: b \sqrt{2}$

$\pink\bigstar$To find:

• The value of a and b

$\red\bigstar$ Solution:

LHS = $\sf\dfrac{3 \: + \: \sqrt{2} }{3 \: - \: \sqrt{2} }$

Let's rationalise the denominator.

$\sf\dfrac{(3 \: + \: \sqrt{2})(3 \: - \: \sqrt{2}) }{(3 \: - \: \sqrt{2})(3 \: - \: \sqrt{2}) } \:$

[By using a² - b² = (a + b)(a - b) and (a - b)² = a² - 2ab + b²]

$\sf = \dfrac{ {(3)}^{2} \: - \: {( \sqrt{2} )}^{2} }{ ({3 \: - \sqrt{2}) }^{2} } \:$

$\sf = \dfrac{ 9 \: - \: 2 }{ {(3)}^{2} \: - \: 2(3)( \sqrt{2}) \: + \: {( \sqrt{2}) }^{2} } \:$

$\sf = \dfrac{ 7 }{ 9\: - \: 6\sqrt{2} \: + \: 2} \:$

[ $\because$ $\sf\sqrt{a} \: \times \: \sqrt{a}$ ]

$\sf = \dfrac{ 7 }{ 11\: - \: 6\sqrt{2} \: }\:$

RHS = $\sf \: a \: + \: b \sqrt{2}$

We know that LHS = RHS [Given]

$\sf \dfrac{ 7 }{ 11\: - \: 6\sqrt{2} \: }\: = \: a \: + \: b \sqrt{2}$

[By comparing both sides ]

$\sf \dfrac{ 7 }{ 11}\: - \dfrac{7}{ 6\sqrt{2} \: }\: = \: a \: + \: b \sqrt{2}$

$\sf \implies \dfrac{ 7 }{ 11}\: = \: a \:$

$\tt And$

$\implies \: - \: \dfrac{7}{ 6\sqrt{2} \: }\: = \: b \sqrt{2}$

$\sf\implies \dfrac{( - 7)( \sqrt{2}) }{6 \sqrt{2} \times \sqrt{2} } = \: b \sqrt{2}$

[ By rationalisation]

$\sf\implies \dfrac{( - 7 \sqrt{2}) }{12 } = \: b \sqrt{2}$

[$\sqrt{a} \: \times \: \sqrt{a}\: = \:a$]

$\sf\implies \dfrac{( - 7 ) }{12 } = \: b$

$\sf( \sqrt{2} \: will \: be \: cancelled \: )$

$\sf \boxed{\therefore \: a \: = \: \dfrac{7}{11} \sf \: and \: \: b \: = \: \dfrac{( - 7)}{12} }$

$\red\bigstar$ Concepts Used:

• Rationalisation of Denominator
• a² - b² = (a + b)(a - b)
• (a - b)² = a² - 2ab + b²
• Equation of variables with their Values
• $\sqrt{a} \: \times \: \sqrt{a}\: = \:a$

$\bigstar$ 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)