Question

0.3 Solve
any three of the followin
1) Rationalize the denominator
1√7+√2
​

Answer 1

Correct Question:

1) Rationalize the Denominator

$\leadsto~~\dfrac{1}{( \sqrt{7} + \sqrt{2} }$

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Explanation

Here, (√7 + √2) is the denominator and it is irrational. As 7 and 2 are not perfect squares, √7 and √2 both are irrational numbers. In the denominator, √7 and √2 are added together. i.e. two irrational numbers are added together, so their sum is also an irrational number. Now, we need to rationalize the denominator i.e. (√7 + √2 ).

Rationalising factor of (√7 + √2 ) is ( √7 - √2 ). So, we can multiply both the numerator and denominator by ( √7 - √2 ). note that the value of fraction will not change as we are multiplying with 1.

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$~~\leadsto~~~ \dfrac{1}{( \sqrt{7} + \sqrt{2} } \times \dfrac{( \sqrt{7} - \sqrt{2}) }{(\sqrt{7} - \sqrt{2)}}$

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

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$~~\leadsto~~~ \dfrac{ \sqrt{7} - \sqrt{2} }{( { \sqrt{7} })^{2} - ( { \sqrt{2} )}^{2} }$

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(√x)² is same as x

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$~~\leadsto~~~ \dfrac{ \sqrt{7} - \sqrt{2} }{7 - 2}$

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$~~\leadsto~~~ \dfrac{ \sqrt{7} - \sqrt{2} }{5}$

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$\bold{\dfrac{ \sqrt{7} - \sqrt{2} }{5}}$ is the form with the rationalized denominator.