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:

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

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

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