Question

$if \: \frac{5 + 2 \: \sqrt[]{3} }{7 + 4 \sqrt{3} } = a + b \sqrt{3} \: \\ \\ then \: a \: \: b \: \: is \: equal \: to \:$
total explanation
no copying!!! leave it if you don't know I don't mind it.
just answer it with correct concept and explanation

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Answer 1

$\red{\bf {Answer:}}$

$\begin{gathered}\\\end{gathered}$

$\underline{\boxed{\sf\purple{a=11, b=-6}}}$

$\begin{gathered}\\\end{gathered}$

$\red{\bf {Solution:}}$

$\begin{gathered}\\\end{gathered}$

$\frac{ (5 + 2√3)}{ (7 + 4√3)}$

$\underline{\boxed{\sf\purple{= a + b√3}}}$

$\begin{gathered}\\\end{gathered}$

Rationalizing the denominator on left-hand-side by multiplying the numerator and denominator with (7 - 4√3),

$\begin{gathered}\\\end{gathered}$

$\frac{(5 + 2√3) (7 - 4√3)}{(7 + 4√3) (7 - 4√3)}$

$\underline{\boxed{\sf\purple{= a + b√3}}}$

$\begin{gathered}\\\end{gathered}$

Multiply term by term the two expressions on numerator of L.H.S. and for the denominator apply the identity (m+n) (m-n) = m² - n² .

$\begin{gathered}\\\end{gathered}$

$\bold{We \:obtain,}$

$\begin{gathered}\\\end{gathered}$

$\frac{(35 - 20√3 + 14√3 - 8.√3.√3)}{[7² - (4√3)²]}$

$\underline{\boxed{\sf\purple{= a + b√3}}}$

$\begin{gathered}\\\end{gathered}$

$\bold{Or, }$

$\begin{gathered}\\\end{gathered}$

$\frac{(35 - 6√3 - 8.3)}{(49 - 48)}$

$\underline{\boxed{\sf\purple{= a + b√3}}}$

$\begin{gathered}\\\end{gathered}$

$\bold{Or, }$

$\begin{gathered}\\\end{gathered}$

$\frac{(35 - 6√3 - 24)}{1}$

$\underline{\boxed{\sf\purple{= a + b√3}}}$

$\begin{gathered}\\\end{gathered} </p><p> </p><p> </p><p></p><p>[tex]\bold{Or, }$

$\underline{\boxed{\sf\purple{11 - 6√3 = a + b√3}}}$

$\begin{gathered}\\\end{gathered}$

Now equate the rational and irrational terms from both sides.

$\begin{gathered}\\\end{gathered}$

$\boxed{\sf{11 = a}}$

$\begin{gathered}\\\end{gathered}$

$\bold{Or, }$

$\begin{gathered}\\\end{gathered}$

$\boxed{\sf{a = 11}}$

$\begin{gathered}\\\end{gathered}$

$\bold{- 6√3 = b√3}$

$\begin{gathered}\\\end{gathered}$

$\underline{\boxed{\sf\purple{⇒ b = -6}}}$

$\begin{gathered}\\\end{gathered}$

Thankyou :)

$\begin{gathered}\\\end{gathered}$

Refer the attachment for better understanding :)

Answer 2

Answer:

Now pls let me thank you and don't give me any more thanks. okey.