How do you simplify $\frac{16}{\sqrt{5} - \sqrt{7}}$ $16/(sqrt [5] - sqrt [7])$ ?

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Answer 1 · 2015-10-16T00:19:47

$- 8 (\sqrt{5} + \sqrt{7})$ $-8(sqrt(5) + sqrt(7))$

To simplify this expression, you have to rationalize the denominator by using its conjugate expression.

For a binomial, you get its conjugate by changing the sign of the second term.

In your case, that would imply having

$sqrt(5) - sqrt(7) -> underbrace(sqrt(5) color(red)(+) sqrt(7))_(color(blue)("conjugate"))$

So, multiply the fraction by $1 = (sqrt(5) + sqrt(7))/(sqrt(5) + sqrt(7))$ to get

$16/(sqrt(5) - sqrt(7)) * (sqrt(5) + sqrt(7))/(sqrt(5) + sqrt(7)) = (16(sqrt(5) + sqrt(7)))/((sqrt(5) - sqrt(7))(sqrt(5)+sqrt(7))$

The denominator is now in the form

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

This means that you can write

$(16(sqrt(5) + sqrt(7)))/( (sqrt(5))^2 - (sqrt(7))^2) = (16(sqrt(5) + sqrt(7)))/(5-7) = color(green)(-8(sqrt(5) + sqrt(7)))$

How do you simplify 16/(sqrt [5] - sqrt [7])16√5−√716/(sqrt [5] - sqrt [7])?