Question

Without using tables or long division find the value of
$\huge \frac{1}{3 - \sqrt{2} }$
√2 = 1.414​

Answer 1

Step-by-step explanation:

1/(3-√2)×3+√2/3+√2

3+√2/9-2

3+√2/7

Answer 2

Basic Concept Used :-

Concept of Rationalization :-

To multiply and divide the numerator and denominator by the opposite sign in denominator to remove the radicals from denominator.

Let's solve the problem now!!

Consider,

$\rm :\longmapsto\:\dfrac{1}{3 - \sqrt{2} }$

On rationalizing the denominator, we get

$\: \sf \: \: = \: \: \dfrac{1}{3 - \sqrt{2} } \times \dfrac{3 + \sqrt{2} }{3 + \sqrt{2} }$

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

$\: \: \: \: \: \: \: \boxed{\red{\sf\: \because \: (x + y)(x - y) = {x}^{2} - {y}^{2} }}$

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

$\: \sf \: \: = \: \: \dfrac{3 + \sqrt{2} }{7}$

$\: \sf \: \: = \: \: \dfrac{3 + 1.414}{7} \: \: \: \: \: \: \: \: \: \: \: \{ \because \: \bf \: \sqrt{2} = 1.414 \}$

$\: \sf \: \: = \: \: \dfrac{4.414}{7}$

$\: \sf \: \: = \: \: 0.6305 \: (approx.)$

More Identities to know:

• (a + b)² = a² + 2ab + b²

• (a - b)² = a² - 2ab + b²

• a² - b² = (a + b)(a - b)

• (a + b)² = (a - b)² + 4ab

• (a - b)² = (a + b)² - 4ab

• (a + b)² + (a - b)² = 2(a² + b²)

• (a + b)³ = a³ + b³ + 3ab(a + b)

• (a - b)³ = a³ - b³ - 3ab(a - b)