numerator: square root of 150x^3 denominator: square root of 6x

To simplify the expression \(\frac{\sqrt{150x^3}}{\sqrt{6x}}\):

### Step 1: Simplify the square roots individually.

1. **Numerator**: \(\sqrt{150x^3}\)
   - Break it down: \(150x^3 = 25 \times 6 \times x^2 \times x\)
   - So, \(\sqrt{150x^3} = \sqrt{25 \times 6 \times x^2 \times x} = \sqrt{25} \times \sqrt{6} \times \sqrt{x^2} \times \sqrt{x}\)
   - This simplifies to \(5 \times \sqrt{6} \times x \times \sqrt{x} = 5x\sqrt{6x}\)

2. **Denominator**: \(\sqrt{6x}\)
   - This is already in its simplest form.

### Step 2: Simplify the expression.

Now, divide the simplified numerator by the simplified denominator:

\[
\frac{5x\sqrt{6x}}{\sqrt{6x}}
\]

### Step 3: Cancel out the common terms.

Since \(\sqrt{6x}\) is present in both the numerator and the denominator, they cancel out:

\[
\frac{5x\sqrt{6x}}{\sqrt{6x}} = 5x
\]

### Final Answer:
The simplified expression is \(5x\).

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Would you like more details or have any questions?

Here are some related questions you might find interesting:
1. How do you simplify expressions involving higher powers under a square root?
2. What are some tips for factoring expressions inside square roots?
3. How does this method change if you have cube roots instead of square roots?
4. How do you simplify a square root when the radicand includes both constants and variables?
5. What happens if the terms inside the square roots are not fully simplified?

**Tip:** Always try to simplify the radicand (the expression inside the square root) as much as possible before taking the square root.

Learn how to simplify the expression (square root of 150x^3) divided by (square root of 6x) step-by-step. This problem involves simplifying square roots and algebraic expressions, suitable for high school students (Grades 9-12).

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