Question

Assuming xx and yy are both positive, write the following expression in simplest radical form.
8, x, cubed, y, square root of, 9, x, to the power 7 , y, cubed, end square root
8x
3
y
9x
7
y
3

​

Answer 1

To simplify the expression \( 8x^3 y \sqrt{9x^7y^3} \), we first simplify the square root portion:

1. Start with the expression inside the square root: \( 9x^7y^3 \).
2. Recognize that \( 9 \) can be simplified as \( 3^2 \) and split \( x^7 \) and \( y^3 \) into their respective perfect squares:
\[
9x^7y^3 = 3^2 \cdot x^6 \cdot x^1 \cdot y^3 = 3^2 \cdot x^6 \cdot y^2 \cdot y
\]
3. Now, take the square root:
\[
\sqrt{9x^7y^3} = \sqrt{3^2} \cdot \sqrt{x^6} \cdot \sqrt{y^2} \cdot \sqrt{xy} = 3 \cdot x^3 \cdot y \cdot \sqrt{xy} = 3x^3y\sqrt{xy}
\]

Now substitute this back into the original expression:
\[
8x^3y \cdot \sqrt{9x^7y^3} = 8x^3y \cdot (3x^3y\sqrt{xy})
\]

Now, multiply the coefficients and variables:
- The coefficients: \( 8 \cdot 3 = 24 \)
- The variable parts: \( x^3 \cdot x^3 = x^{3+3} = x^6 \) and \( y \cdot y = y^{1+1} = y^2 \)

Thus, we can combine everything:
\[
= 24x^6y^2\sqrt{xy}
\]

Therefore, the expression in simplest radical form is:
\[
\boxed{24x^6y^2\sqrt{xy}}
\]