Question

By rationalizing the denominators, simplify (12*sqrt(5))/(2*srqt(5)-4)

Answer 1

\[ \frac{ 12\sqrt 5 }{ 2\sqrt 5-4 }\]

Answer 2

To do this, you have to multiply the numerator & denominator by the "conjugate" of the denominator.

Answer 3

In this case, the conjugate is \[2\sqrt{5}+4\]

Answer 4

When you are looking at a denominator that is a binomial with a square root, that part is critical as any other root involves a much more rigorous process.

Answer 5

Remember, \[(x+y)*(x-y) = x^{2}-y ^{2}\]

Answer 6

So you denominator is (\[2^{2} * \sqrt{5}^{2} - 4^{2}\]

Answer 7

2^2=4
\[\sqrt{5}*\sqrt{5}=5\]
-(4^2) = -(16) = -16

Answer 8

4 * 5 -16 = 4

Answer 9

Now for the numerator.

Answer 10

\[(12\sqrt{5})(2\sqrt{5}-4)=\]

Answer 11

\[(12\sqrt{5}) (2\sqrt{5}-4) =
[(12 * 2) (\sqrt{5} * \sqrt{5})] - [(12 * 4) * \sqrt{5)}] =
[(24 * 5)] - [(48\sqrt{5})] =
120 - 48\sqrt{5}\]

Answer 12

\[(12\sqrt{5}) * (2\sqrt{5} -4) =\]
\[(12*2)*(\sqrt{5}*\sqrt{5}) - ((12*4)*\sqrt{5})) =\]
\[(24) * (5) - (48 * \sqrt{5}) =\]
\[120 - 48\sqrt{5}\]

Answer 13

The final/simplified/rationalized answer is:
\[(120 - 48\sqrt{5}) / 4\]