https://file.aiquickdraw.com/mathbot/file/17254404482518c8xw40p.jpg

To simplify the expression \(\sqrt{72z^5}\), we'll remove all perfect squares from inside the square root.

### Step-by-Step Solution:

1. **Factorize the number under the square root:**
   \[
   72 = 2^3 \times 3^2
   \]
   So, we can write:
   \[
   72z^5 = 2^3 \times 3^2 \times z^5
   \]

2. **Group the perfect squares:**
   \[
   \sqrt{72z^5} = \sqrt{2^3 \times 3^2 \times z^4 \times z}
   \]
   Notice that \(3^2\) and \(z^4\) are perfect squares, so they can be simplified outside the square root.

3. **Simplify the square root:**
   \[
   \sqrt{2^3 \times 3^2 \times z^4 \times z} = 3z^2 \times \sqrt{2 \times 2 \times z} = 6z^2 \times \sqrt{2z}
   \]

So, the simplified form is:
\[
6z^2\sqrt{2z}
\]

Would you like more details or have any questions?

### Related Questions:
1. How do you simplify square roots involving variables?
2. What is the process for factoring numbers to identify perfect squares?
3. Can you explain how to simplify \(\sqrt{50x^3}\) using similar steps?
4. How do you simplify square roots with cube roots inside?
5. What happens when simplifying expressions with fractional exponents?

### Tip:
Always break down the number inside the square root into its prime factors to easily identify and extract perfect squares.

Simplify \(\sqrt{72z^5}\). Remove all perfect squares from inside the square root. Assume \(z\) is positive.

Learn how to simplify the expression \(\sqrt{72z^5}\) by removing all perfect squares from inside the square root. This detailed guide covers the key concepts of simplifying radicals and working with exponents, suitable for 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