### Video Transcript

Simplify negative 36 over root two minus root six by rationalizing the denominator.

We’re given an expression which involves two radicals, or square roots, in the denominator and asked to simplify this expression by rationalizing the denominator. This means we need to rewrite it in a form that has a rational number in the denominator. We do this by multiplying both the numerator and denominator of the fraction by the same value, which is carefully chosen so that its product with the denominator gives a rational value. Because we’re multiplying both the numerator and denominator by the same value, we’re multiplying by a fraction that is equal to one. And so the result of this product is equivalent to the original fraction.

First, we recall that the conjugate of root 𝑎 plus root 𝑏, where 𝑎 and 𝑏 are nonnegative rational numbers, is root 𝑎 minus root 𝑏. Furthermore, the product of a number and its conjugate is rational: root 𝑎 plus root 𝑏 multiplied by root 𝑎 minus root 𝑏 is equal to 𝑎 minus 𝑏. So, we’ll multiply both the numerator and denominator of this quotient by the conjugate of the denominator, which is root two plus root six. Multiplying the two fractions together gives the negative of 36 root two plus 36 root six over two minus six.

The denominator simplifies to negative four. And as four is a factor of 36, we can cancel a factor of negative four throughout. Doing so gives nine root two plus nine root six. Finally, we can factor by nine, and we’ve found that the simplified form of the given quotient is nine multiplied by root two plus root six.