How do you rationalize the denominator and simplify #5/(sqrt[3] + sqrt[5])#?

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Answer 1 · 2015-10-05T05:06:01

#=color(blue)((-5(sqrt3-sqrt5))/2#

Rationalizing involves multiplying the numerator and the denominator of the expression by the conjugate of the denominator.

The conjugate of the denominator is
#sqrt3+sqrt5 = color(blue)(sqrt3-sqrt5#

Rationalizing

#5/(sqrt3+sqrt5)= (5* color(blue)((sqrt3-sqrt5)))/((sqrt3+sqrt5)* color(blue)((sqrt3-sqrt5))#

The denominator can be simplified by applying the property
#(a+b)(a-b) = a^2-b^2#

So,
#(sqrt3+sqrt5)* (sqrt3-sqrt5)=3-5=color(blue)(-2#

The expression then becomes:

#(5sqrt3-5sqrt5)/color(blue)(-2#

#=color(blue)((5(sqrt3-sqrt5))/-2#