QuestionSimplify. Rationalize the denominator. $\frac{-2}{4+\sqrt{15}}$

Studdy Solution

STEP 1

What is this asking? We need to simplify the fraction and get rid of the square root in the bottom. Watch out! Don't forget to distribute the negative sign correctly when simplifying!

STEP 2

1. Rationalize the Denominator
2. Simplify the Expression

STEP 3

Alright, so we've got this fraction $\frac{-2}{4 + \sqrt{15}}$ and we want to get that pesky $\sqrt{15}$ out of the denominator.
How do we do that?
We multiply the top and bottom by the conjugate of the denominator!

STEP 4

The conjugate of $4 + \sqrt{15}$ is $4 - \sqrt{15}$ .
Remember, the conjugate is just the same expression with the sign in the middle flipped.
We're basically using the difference of squares here, which is super useful!

STEP 5

So, let's multiply! $\frac{-2}{4 + \sqrt{15}} \cdot \frac{4 - \sqrt{15}}{4 - \sqrt{15}}$ Remember, multiplying by $\frac{4 - \sqrt{15}}{4 - \sqrt{15}}$ is the same as multiplying by one, so we're not changing the value of the fraction, just how it looks.

STEP 6

Let's multiply the numerators: $-2 \cdot (4 - \sqrt{15}) = -8 + 2\sqrt{15}$ And the denominators: $(4 + \sqrt{15}) \cdot (4 - \sqrt{15}) = 16 - 4\sqrt{15} + 4\sqrt{15} - 15$ Notice how the terms with $\sqrt{15}$ add to zero!
That's exactly what we wanted.
We're left with: $16 - 15 = 1$

STEP 7

So, our fraction now looks like this: $\frac{-8 + 2\sqrt{15}}{1}$ Since dividing by one doesn't change anything, our simplified expression is: $-8 + 2\sqrt{15}$

STEP 8

The simplified form of $\frac{-2}{4+\sqrt{15}}$ , with a rationalized denominator, is $-8 + 2\sqrt{15}$ .

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