Question

Assuming xx and yy are both positive, write the following expression in simplest radical form.
6, y, cubed, square root of, 150, x, to the power 6 , y, to the power 4 , end square root
6y
3

150x
6
y
4

​

Answer 1

To simplify the expression \( 6y^3 \sqrt{150x^6y^4} \), we will follow these steps:

1. **Simplify the square root**: Break down what's under the square root.

\[
150 = 25 \times 6 = 5^2 \times 6
\]

So, we can write:

\[
\sqrt{150} = \sqrt{25 \times 6} = \sqrt{25}\sqrt{6} = 5\sqrt{6}
\]

Next, for \( x^6 \) and \( y^4 \):

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

Putting it together:

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

Thus:

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

2. **Combine the expression**:

Now substitute back into the original expression \( 6y^3 \sqrt{150x^6y^4} \):

\[
6y^3 \cdot (5x^3y^2\sqrt{6}) = 30x^3y^{3+2}\sqrt{6} = 30x^3y^5\sqrt{6}
\]

3. **Final result**:

The simplest radical form of the expression \( 6y^3 \sqrt{150x^6y^4} \) is:

\[
\boxed{30x^3y^5\sqrt{6}}
\]