rationalize the square root of 28x^6y^8 over square root 2z

To rationalize the expression \( \frac{\sqrt{28x^6y^8}}{\sqrt{2z}} \), we will follow these steps:

### Step 1: Simplify each square root
- \( \sqrt{28x^6y^8} \) can be broken down as:
  \[
  \sqrt{28x^6y^8} = \sqrt{28} \cdot \sqrt{x^6} \cdot \sqrt{y^8}
  \]
  - \( \sqrt{28} = \sqrt{4 \times 7} = 2\sqrt{7} \)
  - \( \sqrt{x^6} = x^3 \) (since \( x^6 \) is a perfect square)
  - \( \sqrt{y^8} = y^4 \) (since \( y^8 \) is a perfect square)

Thus, we have:
\[
\sqrt{28x^6y^8} = 2x^3y^4\sqrt{7}
\]

### Step 2: Simplify the denominator
- \( \sqrt{2z} = \sqrt{2} \cdot \sqrt{z} \)

### Step 3: Form the fraction
The expression becomes:
\[
\frac{2x^3y^4\sqrt{7}}{\sqrt{2}\sqrt{z}} = \frac{2x^3y^4\sqrt{7}}{\sqrt{2z}}
\]

### Step 4: Rationalize the denominator
To rationalize the denominator, multiply both the numerator and denominator by \( \sqrt{2z} \):
\[
\frac{2x^3y^4\sqrt{7}}{\sqrt{2z}} \times \frac{\sqrt{2z}}{\sqrt{2z}} = \frac{2x^3y^4\sqrt{14z}}{2z}
\]

### Step 5: Simplify the expression
The final expression is:
\[
\frac{x^3y^4\sqrt{14z}}{z}
\]

This is the rationalized form of the given expression.

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Would you like more details or have any questions about these steps?

Here are five related questions to explore further:
1. How do you rationalize a denominator involving cube roots?
2. How can you simplify more complex expressions under a square root?
3. What are the rules for simplifying exponents under a square root?
4. How do you handle variables under radical expressions when rationalizing?
5. What is the benefit of rationalizing the denominator in mathematical operations?

**Tip:** Always simplify inside the square roots first before attempting to rationalize the denominator!

Explore how to rationalize the denominator of the expression √(28x^6y^8) / √(2z) step-by-step. Learn how to simplify radicals, apply exponent rules, and rationalize the denominator. Ideal for students in Grades 9-11.

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