Question

Rationalise the denominator of each of the following

$\frac{3 - 2 \sqrt{2} }{3 + 2 \sqrt{2} }$
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Answer 1

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Rationalise:

• ${ \sf\cfrac{3 - 2 \sqrt{2} }{3 + 2 \sqrt{2} } }$

On Rationalise the term denominator

${ \implies\sf\cfrac{3 - 2 \sqrt{2} }{3 + 2 \sqrt{2} } \times \cfrac{3 - 2 \sqrt{2} }{3 - 2 \sqrt{2} }}$

${\implies \sf\cfrac{(3 - 2 \sqrt{2}) ^{2} }{ {3}^{2} - (2 \sqrt{2}) ^{2} } }$

${\implies \sf\cfrac{{3}^{2} + (2 \sqrt{2}) ^{2} - 2 \times 3 \times 2 \sqrt{2} }{ 9 - {2}^{2} (\sqrt{2} )^{2} } }$

${\implies \sf\cfrac{9 +8 -12\sqrt{2} }{ 9 - 4 \times 2 } }$

${\implies \sf\cfrac{17 -12\sqrt{2} }{ 9 - 8 } }$

${~~~~~~~~~\underline{\pink{ \boxed{\bf17 -12\sqrt{2} }} } }$

Hope it helpful ❣️

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More Information

• $\\\\\sf(a + b)^{2} = {a}^{2} + {b}^{2} + 2ab$
• $\\\sf(a - b)^{2} = {a}^{2} + {b}^{2} - 2ab$
• $\\\sf(a + b)(a - b) = {a}^{2} - {b}^{2}$
• $\\\sf(a + b + c)^{2} = {a}^{2} + {b}^{2} + {c}^{2}+ 2ab + 2bc + 2ca$
• $\\\sf(a + b) ^{3} = {a}^{3} + b^{3} + 3ab(a + b)$
• $\\\sf(a - b) ^{3} = {a}^{3} - b^{3} - 3ab(a - b)$
• $\\\sf a ^{3} + {b}^{3} = (a + b)(a ^{2} + {b}^{2} - ab)$
• $\\\sf a ^{3} - {b}^{3} = (a - b)(a ^{2} + {b}^{2} + ab)$

Answer 2

Step-by-step explanation:

Given:-

(3-2√2)/(3+2√2)

To find:-

Rationalising the denominator ?

Solution:-

Given that

(3-2√2)/(3+2√2)

Denominator = 3+2√2

We know that

Rationalising factor of a+√b = a-√b

Rationalising factor of 3+2√2 = 3-2√2

On Rationalising the denominator then

=> [(3-2√2)/(3+2√2)]×[(3-2√2)/(3+2√2)]

=> [(3-2√2)(3-2√2)]×[(3+2√2)(3-2√2)]

=> (3-2√2)^2/[(3+2√2)(3-2√2)]

We know that

(a+b)(a-b)=a^2-b^2

Where a = 3 and b=2√2

=> (3-2√2)^2/[(3)^2-(2√2)^2]

=> (3-2√2)^2/(9-8)

=>(3-2√2)^2/1

=>(3-2√2)^2

It is in the form of (a-b)^2

Where a = 3 and b=2√2

We know that

(a-b)^2 = a^2-2ab+b^2

=> (3-2√2)^2

=> 3^2 - 2 (3)(2√2) +(2√2)^2

=> 9-12√2+8

=> 17-12√2

(3-2√2)/(3+2√2) = 17-12√2

Answer:-

(3-2√2)/(3+2√2) = 17-12√2

Used formulae:-

• Rationalising factor of a+√b = a-√b

• (a+b)(a-b)=a^2-b^2

• (a-b)^2 = a^2-2ab+b^2

Rationalising factor:-

The product of two irrational numbers is a rational number then each irrational is called a Rationalising factor of each other.