How do you simplify 4/(sqrt2-5sqrt3)?

1 Answer
George C.
Aug 3, 2017

4/(sqrt(2)-3sqrt(5)) = -4/43(sqrt(2)+3sqrt(5)) = -4/43sqrt(2)-12/43sqrt(5)

Explanation:

The difference of squares identity can be written:

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

Using this with a=sqrt(2) and b=5sqrt(3), we can rationalise the denominator of the given expression by multiplying both numerator and denominator by sqrt(2)+5sqrt(5)...

4/(sqrt(2)-3sqrt(5)) = (4(sqrt(2)+3sqrt(5)))/((sqrt(2)-3sqrt(5))(sqrt(2)+3sqrt(5)))

color(white)(4/(sqrt(2)-3sqrt(5))) = (4(sqrt(2)+3sqrt(5)))/((sqrt(2))^2-(3sqrt(5))^2)

color(white)(4/(sqrt(2)-3sqrt(5))) = (4(sqrt(2)+3sqrt(5)))/(2-45)

color(white)(4/(sqrt(2)-3sqrt(5))) = -4/43(sqrt(2)+3sqrt(5))

color(white)(4/(sqrt(2)-3sqrt(5))) = -4/43sqrt(2)-12/43sqrt(5)

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