How do you rationalize the denominator and simplify $\frac{2}{1 - \sqrt{5}}$ $2/(1-sqrt5)$ ?

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How do you rationalize the denominator and simplify $\frac{2}{1 - \sqrt{5}}$ $2/(1-sqrt5)$ ?

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Answer 1 · 2018-04-28T10:16:58

$- \frac{1 + \sqrt{5}}{2}$ $- (1+sqrt(5) )/2$

Explanation:

To do this we must multiply both numerator and denominator by the denominators conjugate:

$\frac{2}{1 - \sqrt{5}} \times \frac{1 + \sqrt{5}}{1 + \sqrt{5}}$ $2/(1-sqrt(5) ) xx (1+sqrt(5))/(1+sqrt(5))$

As this is just the same as multiplying by 1:

Expanding:

$\Rightarrow \frac{2 (1 + \sqrt{5})}{(1 - \sqrt{5}) (1 + \sqrt{5})}$ $=> (2(1+sqrt(5) )) / (( 1-sqrt(5))(1+sqrt(5))$

$\Rightarrow \frac{2 + 2 \sqrt{5}}{1 - \sqrt{5} + \sqrt{5} - \sqrt{5} \sqrt{5}}$ $=> ( 2 + 2sqrt(5) ) / ( 1 - sqrt(5) + sqrt(5) - sqrt(5)sqrt(5) )$

$\Rightarrow \frac{2 + 2 \sqrt{5}}{- 4}$ $=> ( 2 + 2sqrt(5) ) / ( -4 )$

$\Rightarrow - \frac{1}{2} - \frac{1}{2} \sqrt{5} = - \frac{1 + \sqrt{5}}{2}$ $=> -1/2 -1/2 sqrt(5) = - (1+sqrt(5) )/2$

Answer 2 · 2018-04-28T10:21:30

$- \frac{1}{2} (1 + \sqrt{5})$ $-1/2(1+sqrt5)$

Explanation:

$to rationalise the denominator$ $"to "color(blue)"rationalise the denominator"$

$multiply the numerator/denominator by the conjugate$ $"multiply the numerator/denominator by the "color(blue)"conjugate"$
$of the denominator$ $"of the denominator"$

$the conjugate of 1 - \sqrt{5} is 1 + \sqrt{5}$ $"the conjugate of "1-sqrt5" is "1color(red)(+)sqrt5$

$\Rightarrow \frac{2 (1 + \sqrt{5})}{(1 - \sqrt{5}) (1 + \sqrt{5})}$ $rArr(2(1+sqrt5))/((1-sqrt5)(1+sqrt5))$

$expand the denominator using FOIL$ $"expand the denominator using FOIL"$

$\frac{2 (1 + \sqrt{5})}{- 4} = - \frac{1}{2} (1 + \sqrt{5})$ $(2(1+sqrt5))/(-4)=-1/2(1+sqrt5)$

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How do you rationalize the denominator and simplify $\frac{2}{1 - \sqrt{5}}$ $2/(1-sqrt5)$ ?

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