Math  /  Algebra

QuestionWhich is the rationalized form of the expression πx+5\frac{\sqrt{\pi}}{\sqrt{x}+\sqrt{5}} ? a. xx+5\frac{x}{x+5} c. x5πx5\frac{x-\sqrt{5 \pi}}{x-5} b. x+5πx5\frac{x+\sqrt{5 \pi}}{x-5} d. 5π5\frac{\sqrt{5 \pi}}{5}
Please select the best answer from the choices provided A B C Mark this and return Save and Exit

Studdy Solution

STEP 1

What is this asking? We need to simplify the fraction with square roots by rationalizing the denominator, meaning getting rid of the square roots at the bottom. Watch out! Don't forget to multiply both the top and the bottom by the *conjugate* of the denominator, not just by the denominator itself!

STEP 2

1. Find the conjugate
2. Multiply by the conjugate

STEP 3

The **denominator** of our fraction is x+5\sqrt{x} + \sqrt{5}.
To find its **conjugate**, we simply switch the sign in the middle.
So, the conjugate of x+5\sqrt{x} + \sqrt{5} is x5\sqrt{x} - \sqrt{5}.
It's that easy!

STEP 4

Now, we're going to multiply both the **numerator** and the **denominator** of our original fraction by this **conjugate**.
This is like multiplying by 11, so it doesn't change the *value* of the fraction, just how it looks!

STEP 5

Here's how we do it: πx+5x5x5 \frac{\sqrt{\pi}}{\sqrt{x} + \sqrt{5}} \cdot \frac{\sqrt{x} - \sqrt{5}}{\sqrt{x} - \sqrt{5}}

STEP 6

Let's tackle the **numerator** first.
We multiply π\sqrt{\pi} by x5\sqrt{x} - \sqrt{5} to get πx5π\sqrt{\pi x} - \sqrt{5\pi}.

STEP 7

Now, for the **denominator**.
We multiply x+5\sqrt{x} + \sqrt{5} by x5\sqrt{x} - \sqrt{5}.
This is a difference of squares, which simplifies to (x)2(5)2(\sqrt{x})^2 - (\sqrt{5})^2, which is just x5x - 5.
Awesome!

STEP 8

Putting it all together, we get: πx5πx5 \frac{\sqrt{\pi x} - \sqrt{5\pi}}{x - 5}

STEP 9

Our final simplified, rationalized expression is πx5πx5\frac{\sqrt{\pi x} - \sqrt{5\pi}}{x - 5}.
This matches answer choice C!

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