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What is the quotient of the following expression? State the answer in simplest radical form. Rationalize the denominator if necessary. 36400−−−√ (1 point) Responses 310 3 tenths 36√400√ the fraction with numerator square root of 36 and denominator square root of 400 610 6 tenths 9100

5 answers

To find the quotient of \(\sqrt{36400}\), we start by simplifying the expression.

  1. Factor 36400: We can break down 36400 into its prime factors: \[ 36400 = 364 \times 100 \] Next, we can factor 364 as follows: \[ 364 = 4 \times 91 = 4 \times 7 \times 13 \] Hence, \[ 36400 = 4 \times 7 \times 13 \times 100 = 4 \times 7 \times 13 \times 10^2 \]

  2. Calculate the square root: Now we can find the square root: \[ \sqrt{36400} = \sqrt{4 \times 7 \times 13 \times 100} = \sqrt{4} \times \sqrt{7} \times \sqrt{13} \times \sqrt{100} \] Simplifying further: \[ \sqrt{4} = 2, \quad \sqrt{100} = 10 \] So, \[ \sqrt{36400} = 2 \times 10 \times \sqrt{91} = 20\sqrt{91} \]

Thus, the quotient (the value of \(\sqrt{36400}\)) in simplest radical form is: \[ \boxed{20\sqrt{91}} \]

thats not the answer

Apologies for any confusion. Let's break it down step by step to find \(\sqrt{36400}\) and see if there was an error in the interpretation of the problem.

  1. Identify the square root: First, we need to compute \(\sqrt{36400}\).

  2. Simplify the expression: We can factor 36400 as follows: \[ 36400 = 364 \times 100 \] \[ 100 = 10^2 \]

  3. Calculate square roots of components: We can find \(\sqrt{364}\) and \(\sqrt{100}\): \[ \sqrt{100} = 10 \]

In terms of 364: \[ 364 = 4 \times 91 = 2^2 \times 91 \] Therefore, \[ \sqrt{364} = \sqrt{4 \times 91} = \sqrt{4} \times \sqrt{91} = 2\sqrt{91} \]

  1. Combine the square roots: Thus, \[ \sqrt{36400} = \sqrt{364} \times \sqrt{100} = (2\sqrt{91}) \times 10 = 20\sqrt{91} \]

So, the final answer for \(\sqrt{36400}\) is indeed \(20\sqrt{91}\).

If you are looking for specific values based on options provided, let's clarify them:

  • Responses:
    • 310
    • 3 tenths
    • \( \frac{\sqrt{36}}{\sqrt{400}} \)
    • 610
    • 6 tenths
    • 9100

Checking \( \frac{\sqrt{36}}{\sqrt{400}} \):

Calculating \( \frac{\sqrt{36}}{\sqrt{400}} \): \[ \sqrt{36} = 6, \quad \sqrt{400} = 20, \] \[ \frac{\sqrt{36}}{\sqrt{400}} = \frac{6}{20} = \frac{3}{10} = \text{3 tenths} \]

Comparison:

So, based on the response options, the relevant simplification leads us to:

\(\sqrt{36400}\) does not directly match any of the given options, but:

  • \( \frac{\sqrt{36}}{\sqrt{400}} = \frac{3}{10}\)

Final answer in context: It's possible you meant to compare expressions rather than directly evaluating \(\sqrt{36400}\). If there is any additional clarification on context or specific instructions, please let me know!

which one is the answer

I apologize for the confusion earlier. Given the options you provided, let's summarize the evaluations to see which one corresponds to \(\sqrt{36400}\):

  1. Evaluate \(\sqrt{36400}\):

    • We computed \(\sqrt{36400} = 20\sqrt{91}\) but this choice is not among your options.

  2. Check the options:

    • 310: Not relevant.
    • 3 tenths: Not relevant.
    • \(\frac{\sqrt{36}}{\sqrt{400}}\): This simplifies to \(\frac{6}{20} = \frac{3}{10}\).
    • 610: Not relevant.
    • 6 tenths: This is equivalent to \(\frac{3}{5}\) which isn't relevant here.
    • 9100: Not relevant.

From the options given, if you are looking for a fraction or a decimal representation from that expression \( \frac{\sqrt{36}}{\sqrt{400}} = \frac{3}{10} \), then the closest match is:

Answer: 3 tenths (or \(\frac{3}{10}\)).

Thank you for your patience, and I'm glad we found the correspondence in the response options!