Multiply and remove all perfect squares from inside the square roots. Assume y is positive. 6 times the square root of 15y to the power of 4 times 2 times the square root of 20y squared

Let's multiply the given expression and remove all perfect squares from inside the square roots.

The expression is:
\[
6 \times \sqrt{15y^4} \times 2 \times \sqrt{20y^2}
\]

1. **Multiply the constants**:
   \[
   6 \times 2 = 12
   \]
   So, the expression becomes:
   \[
   12 \times \sqrt{15y^4} \times \sqrt{20y^2}
   \]

2. **Combine the square roots**:
   \[
   12 \times \sqrt{(15y^4) \times (20y^2)}
   \]
   Now, multiply the terms inside the square root:
   \[
   15 \times 20 = 300
   \]
   \[
   y^4 \times y^2 = y^{4+2} = y^6
   \]
   So, the expression becomes:
   \[
   12 \times \sqrt{300y^6}
   \]

3. **Simplify the square root**:
   Let's break down the square root:
   \[
   \sqrt{300y^6} = \sqrt{300} \times \sqrt{y^6}
   \]

   Now, simplify each part:

   - \( \sqrt{300} \) can be broken down into \( \sqrt{100 \times 3} = \sqrt{100} \times \sqrt{3} = 10\sqrt{3} \).
   - \( \sqrt{y^6} = y^3 \) because \( y^6 \) is a perfect square.

   So, the expression simplifies to:
   \[
   12 \times 10 \times y^3 \times \sqrt{3} = 120y^3\sqrt{3}
   \]

The final simplified expression is:
\[
120y^3\sqrt{3}
\]

Would you like more details or have any questions?

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Here are 8 related questions that might interest you:
1. How do you simplify square roots that involve variables?
2. What are the steps for multiplying square roots with coefficients?
3. How do you factor perfect squares out of square roots?
4. Can you explain the laws of exponents when multiplying variables?
5. How do you simplify expressions with multiple square roots?
6. What happens when multiplying square roots with different radicands?
7. How do you determine if a number is a perfect square?
8. Can you simplify expressions with square roots in the denominator?

**Tip:** Always break down the square roots into their prime factors to easily identify and remove perfect squares.

Learn how to multiply and simplify square roots involving variables. This problem involves the multiplication of square roots with coefficients and variables, and simplification using exponentiation rules. Suitable for Grades 9-12.

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