Math, asked by maishameeran, 6 hours ago

5+root3/7-4root3=a+root3 b. IF A AND B ARE RATIONAL NUMBERS FIND THE VALUE OF A AND B​ (step by answer send fast)​

Answers

Answered by user0172
3

Answer:

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Answered by Salmonpanna2022
1

Step-by-step explanation:

 \bf \underline{Given-} \\

 \sf{ \frac{5 +  \sqrt{3} }{7 - 4 \sqrt{3} }  = a + b \sqrt{3} } \\

 \bf \underline{To\: find-} \\

\textsf{the value of a and b = ?}

 \bf \underline{Solution-} \\

\textsf{We have}

 \sf{ \frac{5 +  \sqrt{3} }{7 - 4 \sqrt{3} } } \\

\textsf{The denominator is 7 - 4√3.}\\

\textsf{We know that,}\\

\textsf{Rationalising factor of a - b√c = a + b√c.}

\textsf{So, the rationalising factor of 7 - 4√3 = 7 + 4√3.}\\

\textsf{On rationalising the denominator them}\\

 \sf{ \implies \:  \frac{5 +  \sqrt{3} }{7 - 4 \sqrt{3} }  \times  \frac{7 + 4 \sqrt{3} }{7 + 4 \sqrt{3} } } \\

\sf{ \implies \: \frac{(5 +  \sqrt{3})(7 + 4 \sqrt{3} ) }{(7 - 4 \sqrt{3} )(7 + 4 \sqrt{3}) } } \\

\textsf{Now, comparing the denominator with (a-b)(a+b), we get}\\

 \sf{ \:  \:  \: a = 7 \: and \: b = 4 \sqrt{3}  } \\

\textsf{Using identity (a-b)(a+b) = a² - b², we get}\\

 \sf{ \implies \:  \frac{(5 +  \sqrt{3} )(7 + 4 \sqrt{3} )}{(7 {)}^{2}  - (4 \sqrt{3}  {)}^{2} } } \\

 \sf{ \implies \:  \frac{(5 +  \sqrt{3} )(7 + 4 \sqrt{3} )}{49  - 48} } \\

 \sf{ \implies \:  \frac{(5 +  \sqrt{3} )(7 + 4 \sqrt{3} )}{1} } \\

 \sf{ \implies \:(5 +  \sqrt{3} )(7 + 4 \sqrt{3} )} \\

 \sf{ \implies \:35 + 20 \sqrt{3}  + 7 \sqrt{3} + 4  (\sqrt{3}  {)}^{2}  } \\

 \sf{ \implies \:35 + 27 \sqrt{3}  + 4(3)} \\

 \sf{ \implies \:35 + 27 \sqrt{3} + 12 } \\

 \sf{ \implies \:47 + 27 \sqrt{3} } \\

 \sf{ \therefore \: 47 + 27 \sqrt{3} = a + b \sqrt{3}  } \\

\textsf{On comparing with the value of RHS, we notice that}\\

 \sf{a = 47 \: and \: b = 27 \sqrt{3} } \\

 \bf \underline{Hence, the\: value\: of \: a\:= 47\: and\:b\: = \: 27. } \\

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