How do you rationalize the denominator and simplify #4/(5+sqrt2)#?

2 Answers
Mar 19, 2016

Answer:

#=(20-4sqrt(2))/(23)#

Explanation:

Consider #a^2-b^2=(a-b)(a+b)#

Multiply by 1 but where #1=(5-sqrt(2))/(5-sqrt(2))#

#color(brown)(4/(5+sqrt(2))xx(5-sqrt(2))/(5-sqrt(2)))color(blue)(->(20-4sqrt(2))/(5^2-2)#

#=(20-4sqrt(2))/(23)#

Mar 19, 2016

Answer:

The simplified solution is #(4(5 - sqrt2))/23#

Explanation:

To rationalize the denominator you need to multiply it by a term that will eliminate the surd
You must then multiply the numerator by the same term to keep the value of the fraction the same.
So multiply by #(5 - sqrt2)/(5 - sqrt 2)#
this gives

#(4(5 - sqrt2))/( (5 + sqrt2)(5 - sqrt2)# = #(4(5 - sqrt2))/(25 -2)#

So the solution is #(4(5 - sqrt2))/ 23#