Question

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

Answer 1

`=(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)`

Answer 2

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`