Question

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

Answer 1

thats not the answer

Answer 2

which one is the answer

Answer 3

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:
\[
\boxed{20\sqrt{91}}
\]

Answer 4

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,
\[
\sqrt{364} = \sqrt{4 \times 91} = \sqrt{4} \times \sqrt{91} = 2\sqrt{91}
\]

4. **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!

Answer 5

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!