simplify 5, square root of 27x multiplied by square root of 75

To simplify the expression \( 5 \times \sqrt{27x} \times \sqrt{75} \), we can follow these steps:

1. **Simplify the square roots:**
   - \( \sqrt{27x} = \sqrt{27} \times \sqrt{x} \)
   - \( \sqrt{27} = \sqrt{9 \times 3} = \sqrt{9} \times \sqrt{3} = 3\sqrt{3} \)
   - So, \( \sqrt{27x} = 3\sqrt{3} \times \sqrt{x} = 3\sqrt{3x} \)
   - \( \sqrt{75} = \sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3} = 5\sqrt{3} \)

2. **Substitute these back into the expression:**
   \[
   5 \times (3\sqrt{3x}) \times (5\sqrt{3})
   \]

3. **Combine the constants and square roots:**
   \[
   5 \times 3 \times 5 \times \sqrt{3x} \times \sqrt{3} = 75 \times \sqrt{3x} \times \sqrt{3}
   \]

4. **Simplify the square root multiplication:**
   \[
   \sqrt{3x} \times \sqrt{3} = \sqrt{3 \times 3x} = \sqrt{9x} = 3\sqrt{x}
   \]

5. **Multiply the constants:**
   \[
   75 \times 3\sqrt{x} = 225\sqrt{x}
   \]

Thus, the simplified expression is \( \boxed{225\sqrt{x}} \).

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Here are 5 related questions you might find useful:
1. How would you simplify \( \sqrt{45x^2} \times \sqrt{20} \)?
2. Can you simplify \( 7 \times \sqrt{12x} \times \sqrt{18} \)?
3. What is the simplified form of \( \sqrt{50} \times \sqrt{2x} \)?
4. How do you simplify \( 2\sqrt{75y} \times \sqrt{8} \)?
5. What are the steps to simplify \( \sqrt{48} \times \sqrt{3x} \)?

**Tip:** When simplifying square roots, always look for the largest perfect square factor to break down the square root into its simplest form.

Learn how to simplify the expression 5 times the square root of 27x times the square root of 75 step-by-step. This problem involves simplifying square roots and multiplying them, covering key concepts in algebra and arithmetic. Suitable for Grades 9-12 students.

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