https://file.aiquickdraw.com/mathbot/file/1729486962589f3qf2rq1.jpg

The factor tree for \(1,764\) is clearly shown in the image, and the prime factorization of \(1,764\) can be obtained by following it.

From the image, the factor tree breaks down as follows:

\[
1,764 = 2^2 \times 3^2 \times 7^2
\]

Now, to simplify \( \sqrt{1,764} \):

\[
\sqrt{1,764} = \sqrt{2^2 \times 3^2 \times 7^2}
\]

Since all the exponents are even, you can take the square root of each factor:

\[
\sqrt{1,764} = 2 \times 3 \times 7 = 42
\]

Thus, the simplest form of \( \sqrt{1,764} \) is **42**.

### Answer: \( 42 \)

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Do you need any further clarification? Here are some related questions:

1. What is the process for finding the prime factorization of a number?
2. Can every number be simplified using a factor tree?
3. How would you handle a number with more complex prime factorizations?
4. Why does the square root of a number with all even exponents simplify easily?
5. How can you recognize perfect squares when factoring?

**Tip**: Always look for pairs of prime factors when simplifying square roots—it indicates that they can be pulled out of the radical!

Learn how to simplify √1,764 using prime factorization and a factor tree. The number 1,764 is factored as 2^2 × 3^2 × 7^2, and its square root simplifies to 42. This problem demonstrates key concepts in number theory, including the fundamental theorem of arithmetic.

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