Question

Which number is equivalent to: 610−−√⋅39–√ ?(1 point) Responses 1809–√ 180 square root of 9 990−−√ 9 square root of 90 5410−−√ 54 square root of 10 919−−√

Answer 1

To simplify the expression \( \sqrt{610} \cdot \sqrt{39} \), we can use the property of square roots that states \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \).

So we have:

\[
\sqrt{610} \cdot \sqrt{39} = \sqrt{610 \cdot 39}
\]

Now we calculate \( 610 \cdot 39 \):

\[
610 \cdot 39 = 23790
\]

Thus, we have:

\[
\sqrt{610} \cdot \sqrt{39} = \sqrt{23790}
\]

Now, we should see if \( \sqrt{23790} \) can be simplified further by factoring it. Let's break down \( 23790 \):

- \( 23790 \) can be divided by \( 10 \):
\[
23790 = 10 \cdot 2379
\]

- Then, we can factor \( 2379 \):
- \( 2379 = 3 \cdot 793 \) (it's divisible by \( 3 \)).
- \( 793 \) is a prime number.

So:

\[
23790 = 2 \cdot 3 \cdot 5 \cdot 793
\]

This results in:

\[
\sqrt{23790} = \sqrt{2 \cdot 3 \cdot 5 \cdot 793}
\]

To see if it matches any of the options provided, we note that:

- \( \sqrt{1809} \) has factors that can allow further simplification, we try to find the products of the given responses.

Let's check the given responses one by one:

1. \( 1809^{1/2} \)
2. \( 9 \cdot \sqrt{90} \) (which equals \( 9 \cdot \sqrt{9 \cdot 10} = 27 \cdot \sqrt{10} \), not equivalent.)
3. \( 990^{1/2} \)
4. \( 54\cdot \sqrt{10} \)
5. \( 9^{1/2} \)

It looks like \( 54 \sqrt{10} = \sqrt{54^2 \cdot 10} = \sqrt{2904}\), not equal.

The true conversion of trying all possible would be to regard checking them over numerical square roots.

As a final note, the choice generator responses collectively lead toward \( 1809^{1/2}\) showing valid factors.

So:

\[
\sqrt{23790} = 1809
\]

As such, the number that is equivalent to your original expression is:

**Response: \(1809^{1/2}\)**