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Math  /  Algebra

QuestionRationalize the denominator and simplify: 99+11=\frac{9}{9+\sqrt{11}}=

Studdy Solution

STEP 1

What is this asking? We need to rewrite the fraction so that there's no square root in the bottom part, and then make it as simple as possible! Watch out! Don't forget to multiply both the top and bottom by the *conjugate* and not just the square root itself.

STEP 2

1. Find the conjugate
2. Multiply by the conjugate
3. Simplify

STEP 3

The **conjugate** of 9+119 + \sqrt{11} is 9−119 - \sqrt{11}.
We switch the sign in the middle!
We do this because multiplying by the conjugate will help us get rid of the square root in the denominator.

STEP 4

We **multiply** both the **numerator** and the **denominator** of our fraction by the conjugate we just found.
This is like multiplying by one, so it doesn't change the *value* of the fraction, just how it looks!

STEP 5

99+11⋅9−119−11 \frac{9}{9 + \sqrt{11}} \cdot \frac{9 - \sqrt{11}}{9 - \sqrt{11}}

STEP 6

Let's handle the **numerator** first: 9⋅(9−11)=81−911 9 \cdot (9 - \sqrt{11}) = 81 - 9\sqrt{11}

STEP 7

Now, the **denominator**: (9+11)⋅(9−11)=9⋅9−911+911−1111 (9 + \sqrt{11}) \cdot (9 - \sqrt{11}) = 9 \cdot 9 - 9\sqrt{11} + 9\sqrt{11} - \sqrt{11}\sqrt{11} Notice how the terms with 11\sqrt{11} add to zero!
This is exactly why we used the conjugate!

STEP 8

We simplify the **denominator**: 81−11=70 81 - 11 = 70

STEP 9

Putting it all together: 81−91170 \frac{81 - 9\sqrt{11}}{70}

STEP 10

We check if we can simplify the fraction further.
Since **81**, **9**, and **70** don't share any common factors other than one, this fraction is in its simplest form!

STEP 11

81−91170 \frac{81 - 9\sqrt{11}}{70}

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