Write square root of, minus, 175, end square root 
−175
​
  in simplest radical form.

The expression you're looking to simplify is:

\[
\sqrt{-175}
\]

Since the square root of a negative number introduces an imaginary number, we can rewrite this as:

\[
\sqrt{-175} = \sqrt{-1} \times \sqrt{175} = i \times \sqrt{175}
\]

Now, simplify \( \sqrt{175} \). Break 175 into its prime factors:

\[
175 = 25 \times 7 = 5^2 \times 7
\]

Thus, the square root of 175 is:

\[
\sqrt{175} = \sqrt{5^2 \times 7} = 5\sqrt{7}
\]

Finally, putting everything together:

\[
\sqrt{-175} = i \times 5\sqrt{7} = 5i\sqrt{7}
\]

So, the simplest radical form of \( \sqrt{-175} \) is:

\[
5i\sqrt{7}
\]

Would you like further clarification on any steps?

Here are 5 related questions:
1. How do you simplify other square roots with negative radicands?
2. What are imaginary numbers and how are they used?
3. How do you factor numbers under square roots?
4. What is the relationship between square roots and prime factorization?
5. How do you handle square roots of fractions?

**Tip:** Remember that the square root of a negative number always includes the imaginary unit \(i\).

Write square root of, minus, 175, end square root −175 in simplest radical form.

Learn how to simplify the square root of -175 using imaginary numbers. This detailed step-by-step guide breaks down key mathematical concepts, including radical expressions, prime factorization, and imaginary numbers. Suitable for high school students in Grades 9-12.

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