https://file.aiquickdraw.com/mathbot/file/17303939772135babtmqp.jpg

To simplify the expression \(\sqrt{600p^9q^{16}}\), let's break it down step-by-step:

1. **Separate the constants and variables**: 
   \[
   \sqrt{600p^9q^{16}} = \sqrt{600} \cdot \sqrt{p^9} \cdot \sqrt{q^{16}}
   \]

2. **Simplify the square root of 600**:
   - Prime factorize 600: \(600 = 2^3 \cdot 3 \cdot 5^2\).
   - Take the square root: \(\sqrt{600} = \sqrt{(2^2 \cdot 5^2) \cdot (2 \cdot 3)} = 10\sqrt{6}\).

   So, \(\sqrt{600} = 10\sqrt{6}\).

3. **Simplify \(\sqrt{p^9}\)**:
   - Rewrite \(p^9\) as \(p^8 \cdot p\).
   - \(\sqrt{p^8 \cdot p} = \sqrt{p^8} \cdot \sqrt{p} = p^4 \cdot \sqrt{p}\).

   So, \(\sqrt{p^9} = p^4 \sqrt{p}\).

4. **Simplify \(\sqrt{q^{16}}\)**:
   - Since \(16\) is an even power, we can directly take the square root: \(\sqrt{q^{16}} = q^8\).

5. **Combine all parts**:
   \[
   \sqrt{600p^9q^{16}} = 10\sqrt{6} \cdot p^4 \sqrt{p} \cdot q^8 = 10p^4q^8\sqrt{6p}
   \]

Therefore, the expression in simplest radical form is:
\[
10p^4q^8\sqrt{6p}
\]

### Answer:
The correct answer is:
- **Option (D)**: \(10p^4q^8\sqrt{6p}\)

Would you like further details on any part of the solution?

---

Here are some related questions to practice:

1. How would you simplify \(\sqrt{150p^{5}q^{10}}\)?
2. How can you simplify the square root of a product like \(\sqrt{50 \cdot 18}\)?
3. What is the square root of \(x^{12}y^{10}\)?
4. How would you simplify \(\sqrt{24x^3y^5}\)?
5. How can you express \(\sqrt{500}\) in simplest form?

**Tip:** Remember, when simplifying square roots with exponents, separate the expression into perfect squares and non-perfect squares to make the process easier.

Write the expression in simplest radical form: \( \sqrt{600p^9q^{16}} \)

Learn how to simplify the radical expression \( \sqrt{600p^9q^{16}} \) by breaking it down step-by-step. This guide covers the square root of constants and variables with exponents, showing how to rewrite in simplest radical form. Suitable for high school students in Grades 9-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation