Math, asked by varsha145, 1 year ago

Find a and b= √11-√7by√11+√7=a-b√77

Answers

Answered by abhi569
30
Solution:-

 \frac{ \sqrt{11}- \sqrt{7}  }{ \sqrt{11} +  \sqrt{7} } = a-b \sqrt{77}

Now,

By Rationalization,

 \frac{ \sqrt{11} -  \sqrt{7} }{ \sqrt{11} +  \sqrt{7} } × \frac{ \sqrt{11}- \sqrt{7}  }{ \sqrt{11}- \sqrt{7}  }

Now,
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On Denominator

By Formula :-(a+b)(a-b) = a^2 - b^2

On  Nominator,

By Formula :- a^2 + b^2+2ab = (a+b)^2
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Then,

 \frac{( \sqrt{11}- \sqrt{7})^2  }{( \sqrt{11})^2 -( \sqrt{ 7 })^2  }


 \frac{11+7-2 \sqrt{77} }{11-7}


 \frac{18- 2\sqrt{77} }{4} a-b \sqrt{77}

 \frac{18}{4} -  \frac{2 \sqrt{77} }{4}

Then,

a =  \frac{18}{4}

a= \frac{9}{2}  

b \sqrt{77}  \frac{2 \sqrt{77} }{4} = \frac{ \sqrt{77}}{2}

b =  \frac{1}{2}



i hope this will help you


-by ABHAY

abhi569: please go back and see it again
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abhi569: now, see it
varsha145: it is wrong answer
abhi569: waity
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varsha145: sorry it is correct
abhi569: its ok..............it is the edited asnwer
Answered by HarishAS
20
Hi friend, Harish here.

Here is your answer:

Given that,

 \frac{\sqrt{11}- \sqrt{7}  }{\sqrt{11}+\sqrt{7}} =  a- b \sqrt{77}

To find,  

The value of a & b.

Solution:

First we must rationalize the denominator of the given number by multiplying and dividing with it's conjugate.

Conjugate of denominator is √11 - √7.

Then,

  \frac{\sqrt{11}- \sqrt{7} }{\sqrt{11}+ \sqrt{7} } \times  \frac{\sqrt{11}- \sqrt{7} }{\sqrt{11}- \sqrt{7} } =  \frac{(\sqrt{11}- \sqrt{7} )^{2} }{(\sqrt{11}- \sqrt{7})(\sqrt{11}+ \sqrt{7})  }

Here in the denominator to multiply we can use the identity:

(x+y)(x-y) = x² - y².

Then,

(√11 + √7) (√11 - √7) = (√11)² - (√7)² = 11 - 7 = 4

So,

 \frac{(\sqrt{11}- \sqrt{7} )^{2} }{(\sqrt{11}- \sqrt{7})(\sqrt{11}+ \sqrt{7}) } =  \frac{11+7-2( \sqrt{11})( \sqrt{7})  }{4} =  \frac{18 - 2 \sqrt{77} }{4}

⇒  \frac{18 - 2 \sqrt{77} }{4}  = a-b \sqrt{77}

⇒   \frac{18}{4} -  \frac{2 \sqrt{77} }{4} = a-b \sqrt{77}

Now by comparing the above equation , We get:

a= \frac{18}{4} =  \frac{9}{2}   : b=  \frac{2}{4} =  \frac{1}{2}

Hence these are the values of a & b.
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Hope my answer is helpful to you

abhi569: the last step
HarishAS: U did it wrong bro, U check again.
abhi569: u should see me new answer
abhi569: edited answer
HarishAS: Yeah, Edited answer is also wrong.
HarishAS: Check again
abhi569: wait
HarishAS: Don't include root 77
abhi569: yeah
abhi569: thanks
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