How do you rationalize the denominator and simplify $(3-sqrt5)/(4+sqrt5)$ ?

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Answer 1 · 2016-03-17T01:23:43

$(17-7sqrt(5))/11$

$1$ . Start by multiplying the numerator and denominator by the conjugate of the fraction's denominator, $4-sqrt(5)$ .

$(3-sqrt(5))/(4+sqrt(5))$

$=(3-sqrt(5))/(4+sqrt(5))((4-sqrt(5))/(4-sqrt(5)))$

$2$ . Simplify the numerator.

$=(12-3sqrt(5)-4sqrt(5)+5)/(4+sqrt(5))(1/(4-sqrt(5)))$

$=(17-7sqrt(5))/(4+sqrt(5))(1/(4-sqrt(5)))$

$3$ . Simplify the denominator. Note that it contains a difference of squares $(color(red)(a^2-b^2=(a+b)(a-b)))$ .

$=(17-7sqrt(5))/(16-5)$

$=color(green)(|bar(ul(color(white)(a/a)(17-7sqrt(5))/(11)color(white)(a/a)|)))$

How do you rationalize the denominator and simplify (3-sqrt5)/(4+sqrt5)(3-sqrt5)/(4+sqrt5)?