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Math  /  Algebra

QuestionSimplify completely. 6842\frac{\sqrt{6} \sqrt{84}}{\sqrt{2}} A. 252 B. 252\sqrt{252} C. 767 \sqrt{6} D. 676 \sqrt{7} E. 36736 \sqrt{7}

Studdy Solution

STEP 1

What is this asking? We need to simplify a big square root fraction and pick the right multiple choice answer. Watch out! Don't forget to simplify completely!
It's easy to miss a step and get the wrong answer.

STEP 2

1. Combine the square roots in the numerator.
2. Simplify the combined square root.
3. Divide the square roots.
4. Simplify the final expression.

STEP 3

We can **combine** the square roots in the numerator by **multiplying** the numbers inside: 6⋅84=6⋅84=504 \sqrt{6} \cdot \sqrt{84} = \sqrt{6 \cdot 84} = \sqrt{504} So, our expression becomes: 5042 \frac{\sqrt{504}}{\sqrt{2}} Why did we do this?
It helps us simplify the expression by getting everything under one big square root!

STEP 4

Let's **break down** 504\sqrt{504} by finding the **prime factorization** of 504: 504=2⋅252=2⋅2⋅126=2⋅2⋅2⋅63=2⋅2⋅2⋅3⋅3⋅7=23⋅32⋅7 504 = 2 \cdot 252 = 2 \cdot 2 \cdot 126 = 2 \cdot 2 \cdot 2 \cdot 63 = 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 7 = 2^3 \cdot 3^2 \cdot 7 Now we can **rewrite** 504\sqrt{504} as: 504=23⋅32⋅7 \sqrt{504} = \sqrt{2^3 \cdot 3^2 \cdot 7}

STEP 5

Remember, the square root of a number squared is just the number itself!
So, we can simplify like this: 504=23⋅32⋅7=22⋅2⋅32⋅7=22⋅32⋅2⋅7=2⋅3⋅14=614 \sqrt{504} = \sqrt{2^3 \cdot 3^2 \cdot 7} = \sqrt{2^2 \cdot 2 \cdot 3^2 \cdot 7} = \sqrt{2^2} \cdot \sqrt{3^2} \cdot \sqrt{2 \cdot 7} = 2 \cdot 3 \cdot \sqrt{14} = 6\sqrt{14}

STEP 6

Now, our expression looks like this: 6142 \frac{6\sqrt{14}}{\sqrt{2}}

STEP 7

We can **divide** the square roots by dividing the numbers inside: 142=142=7 \frac{\sqrt{14}}{\sqrt{2}} = \sqrt{\frac{14}{2}} = \sqrt{7} So, our expression becomes: 67 6\sqrt{7}

STEP 8

Our expression is 676\sqrt{7}.
Since 7 is a prime number, we can't simplify any further!
Awesome!

STEP 9

Our final answer is 676\sqrt{7}, which is answer choice D!

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