https://file.aiquickdraw.com/mathbot/file/1725415608545cqux4dnc.jpg

The expression given in the image is:

\[
6y^3 \sqrt{150x^2 y^4}
\]

We need to simplify this expression, assuming \(x\) and \(y\) are both positive.

### Step 1: Simplify the expression under the square root
The expression under the square root is \(150x^2 y^4\). We can factor this as:

\[
150x^2 y^4 = 150 \cdot x^2 \cdot y^4
\]

Next, factor 150 into its prime factors:

\[
150 = 2 \times 3 \times 5^2
\]

So, we can write:

\[
\sqrt{150x^2 y^4} = \sqrt{2 \times 3 \times 5^2 \times x^2 \times y^4}
\]

### Step 2: Take the square root of each factor
The square root of a product is the product of the square roots of each factor:

\[
\sqrt{150x^2 y^4} = \sqrt{2} \times \sqrt{3} \times \sqrt{5^2} \times \sqrt{x^2} \times \sqrt{y^4}
\]

Simplifying this, we have:

\[
\sqrt{150x^2 y^4} = \sqrt{6} \times 5 \times x \times y^2
\]

So:

\[
\sqrt{150x^2 y^4} = 5x y^2 \sqrt{6}
\]

### Step 3: Substitute back into the original expression
Substituting this back into the original expression:

\[
6y^3 \sqrt{150x^2 y^4} = 6y^3 \times 5xy^2 \times \sqrt{6}
\]

### Step 4: Simplify the expression
Multiply the constants and combine the like terms:

\[
6 \times 5 \times y^3 \times x \times y^2 \times \sqrt{6} = 30x y^5 \sqrt{6}
\]

### Final Answer
The simplified expression is:

\[
\boxed{30x y^5 \sqrt{6}}
\]

Would you like further details or have any questions?

### Related Questions:
1. How do you simplify expressions with different powers under a square root?
2. What is the process of rationalizing a denominator involving square roots?
3. How do you factor large numbers like 150 to simplify square roots?
4. What are the properties of exponents that are useful when simplifying square roots?
5. How can you verify if an expression is fully simplified?

### Tip:
When simplifying square roots, always look for perfect square factors under the root to simplify the expression quickly.

Assuming x and y are both positive, write the following expression in simplest radical form: 6y^3 √(150x^2 y^4)

Learn how to simplify the expression 6y^3 √(150x^2 y^4) step by step. This problem explores the properties of square roots and the process of simplifying radicals, suitable for students in Grades 9-11.

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