What is the quotient of the following expression? State the answer in simplest radical form. Rationalize the denominator if necessary. 980√490√ (1 point) Responses 2/√3 the fraction with numerator square root of 2 and denominator 3 2√2 the fraction with numerator square root of 2 and denominator 2 /32√ the fraction with numerator 3 and denominator square root of 2 32√2

Answer 1

To find the quotient of the expression \( \frac{980 \sqrt{490}}{1} \), let’s first simplify \( \sqrt{490} \).

1. Simplifying \(\sqrt{490}\): \[ \sqrt{490} = \sqrt{49 \times 10} = \sqrt{49} \times \sqrt{10} = 7\sqrt{10} \]

2. Substituting back into the original expression: \[ 980 \sqrt{490} = 980 \times 7 \sqrt{10} = 6860 \sqrt{10} \]

Now, if we assume you're looking for a quotient where a denominator is implied, but I see no denominator here, we can't perform any division. However, since it seems there are provided responses, let’s assume there might be a common denominator encountered in context.

Rationalizing Denominator: For the purpose of rationalizing, let's put this in a form: If we want the result in terms of \( \sqrt{3} \) or another number, let's say we're working with finding a fraction in terms of \( \sqrt{3} \).

If we constructed an expression like: \[ \frac{6860\sqrt{10}}{\sqrt{3}} \] To rationalize, we multiply numerator and denominator by \( \sqrt{3} \): \[ \frac{6860\sqrt{10} \cdot \sqrt{3}}{\sqrt{3} \cdot \sqrt{3}} = \frac{6860 \sqrt{30}}{3} \]

But, since without specific direction we cannot know what to divide by, the original:

\[ 980 \sqrt{490} \rightarrow 6860 \sqrt{10} \]

is obtained.

If there is a need to consider anything specific among given responses, let’s work through them for clarity:

• \( 2/\sqrt{3} \)
• \( 2\sqrt{2} \)
• \( 3/\sqrt{2} \)
• \( 32\sqrt{2} \)

However, we cannot directly conclude without specifics on those rates.

If instead, you simply want the clean numerator and radical form of what came out, we maintain that value unless as per your additional point, we address relevance in scenarios.

But as far as we simplified, it holds to: \[ 6860\sqrt{10} \] Or its rational form based on required factors. Thus, please specify if there's any division requested!

Answer 2

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Answer 3

Apologies for the confusion earlier! Let's clarify the problem you provided and arrive at the correct answer based on the stated options.

You asked for the quotient of \( 980\sqrt{490} \). We'll simplify step-by-step to match the expected format outlined in your original question.

Step 1: Simplify \( \sqrt{490} \)

\[ \sqrt{490} = \sqrt{49 \times 10} = 7\sqrt{10} \]

Step 2: Substitute back into the expression

Now substituting back into the expression gives:

\[ 980\sqrt{490} = 980 \times 7\sqrt{10} = 6860\sqrt{10} \]

Now, we need to look at the options you provided and see if we can relate our simplified form to any of them.

Identifying the options:

1. \( \frac{2}{\sqrt{3}} \)
2. \( 2\sqrt{2} \)
3. \( \frac{3}{\sqrt{2}} \)
4. \( 32\sqrt{2} \)

None of these directly correspond with \( 6860\sqrt{10} \).

If you were looking for a quotient involving a division by another radical or number (as per the typical setup of fractions or given responses), it wasn't explicitly mentioned.

Given the options focus on simpler radicals, it’s most likely:

1. We might be meant to compare dimensions, but more likely the final expression can be simply \( 32\sqrt{2} \) or another related assessment.

Conclusion: Since the response context was somewhat broad, the essential direct quotient remains: \[ 6860\sqrt{10} \]

But from your options and raw divisions potentially align with \( 32\sqrt{2} \).

Can you confirm or specify the operations on what numbers you'd need precisely? Your clarity will direct the proceedings directly. Thank you!