Question

simplify by rationalising
denominator
(a) √5-2/5+2 - √5+2/√5-2
​

Answer 1

GIVEN :-

• (√5 - 2)/(√5 + 2) - (√5 + 2)/(√5 - 2)

TO FIND :-

• Value of (√5 - 2)/(√5 + 2) - (√5 + 2)/(√5 - 2).

SOLUTION :-

$: \implies \displaystyle \sf \: \frac{ \sqrt{5} - 2}{ \sqrt{5} + 2} - \frac{ \sqrt{5} + 2}{ \sqrt{5} - 2 }$

$: \implies \displaystyle \sf \: \frac{ (\sqrt{5} - 2)( \sqrt{5} - 2) }{( \sqrt{5} + 2)( \sqrt{5} - 2) } - \frac{ (\sqrt{5} + 2)( \sqrt{5} + 2)}{( \sqrt{5} - 2)( \sqrt{5} + 2) }$

$: \implies \displaystyle \sf \: \frac{( \sqrt{5} - 2) ^{2} }{( \sqrt{5}) ^{2} - (2) ^{2} } - \frac{( \sqrt{5} + 2) ^{2} }{( \sqrt{5) ^{2} } - (2) ^{2} }$

$: \implies \displaystyle \sf \: \frac{( \sqrt{5}) ^{2} + (2) ^{2} - 2 \times \sqrt{5} \times 2 }{5 - 4} - \frac{( \sqrt{5}) ^{2} + (2) ^{2} + 2 \times \sqrt{5} \times 2 }{5 - 4}$

$: \implies \displaystyle \sf \: \frac{5 + 4- 4 \sqrt{5} }{1} - \frac{5 + 4 + 4 \sqrt{5} }{1}$

$: \implies \displaystyle \sf \: \frac{5 + 4- 4 \sqrt{5} - ( 5 + 4 + 4 \sqrt{5}) }{1}$

$: \implies \displaystyle \sf \: 5 + 4 - 4 \sqrt{5} - 5 - 4 - 4 \sqrt{5}$

$: \implies \displaystyle \sf \: - 4 \sqrt{5} - 4 \sqrt{5}$

$: \implies \underline{ \boxed{\displaystyle \sf \: \frac{ \sqrt{5} - 2}{ \sqrt{5} + 2} - \frac{ \sqrt{5} + 2}{ \sqrt{5} - 2 } = - 8 \sqrt{5} }}$

Answer 2

Given

• √5-2 / √5+2 - √5+2 / √5-2

We Find

• Value of given Equation

According to the question

$\begin{gathered} \implies \displaystyle \sf \: \frac{ (\sqrt{5} - 2)( \sqrt{5} - 2) }{( \sqrt{5} + 2)( \sqrt{5} - 2) } - \frac{ (\sqrt{5} + 2)( \sqrt{5} + 2)}{( \sqrt{5} - 2)( \sqrt{5} + 2) } \\ \\ \\\end{gathered}$

$\begin{gathered} \implies \displaystyle \sf \: \frac{( \sqrt{5} - 2) ^{2} }{( \sqrt{5}) ^{2} - (2) ^{2} } - \frac{( \sqrt{5} + 2) ^{2} }{( \sqrt{5) ^{2} } - (2) ^{2} } \\ \\ \\\end{gathered}$

$\begin{gathered} \implies \displaystyle \sf \: \frac{( \sqrt{5}) ^{2} + (2) ^{2} - 2 \times \sqrt{5} \times 2 }{5 - 4} - \frac{( \sqrt{5}) ^{2} + (2) ^{2} + 2 \times \sqrt{5} \times 2 }{5 - 4} \\ \\ \\\end{gathered}$

$\begin{gathered} \implies \displaystyle \sf \: \frac{5 + 4- 4 \sqrt{5} }{1} - \frac{5 + 4 + 4 \sqrt{5} }{1} \\ \\ \\\end{gathered}$

$\begin{gathered} \implies \displaystyle \sf \: \frac{5 + 4- 4 \sqrt{5} - ( 5 + 4 + 4 \sqrt{5}) }{1} \\ \\ \\\end{gathered}$

$\begin{gathered} \implies \displaystyle \sf \: 5 + 4 - 4 \sqrt{5} - 5 - 4 - 4 \sqrt{5} \\ \\ \\\end{gathered}$

$\begin{gathered} \implies \displaystyle \sf \: - 4 \sqrt{5} - 4 \sqrt{5} \\ \\ \\\end{gathered}$

$\begin{gathered} \red\sf \: = - 8 \sqrt{5}\\ \\ \\\end{gathered}</p><p></p><p>$