Question
Assuming xx and yy are both positive, write the following expression in simplest radical form.
square root of, 18, x, to the power 6 , y, to the power 5 , end square root
18x
6
y
5
square root of, 18, x, to the power 6 , y, to the power 5 , end square root
18x
6
y
5
Answers
GPT-4o mini
To simplify the expression \(\sqrt{18x^6y^5}\), we can break it down as follows:
1. **Simplify the numerical part:**
- The prime factorization of 18 is \(2 \times 3^2\).
- Therefore, \(\sqrt{18} = \sqrt{2 \times 3^2} = 3\sqrt{2}\).
2. **Simplify the variable part:**
- For \(x^6\), we can extract squares: \(\sqrt{x^6} = x^{6/2} = x^3\).
- For \(y^5\), we can extract squares as well: \(\sqrt{y^5} = \sqrt{y^4 \times y} = \sqrt{y^4} \cdot \sqrt{y} = y^{4/2} \cdot \sqrt{y} = y^2\sqrt{y}\).
3. **Combine all parts:**
- Putting it all together, we get:
\[
\sqrt{18x^6y^5} = \sqrt{18} \cdot \sqrt{x^6} \cdot \sqrt{y^5} = 3\sqrt{2} \cdot x^3 \cdot (y^2\sqrt{y}) = 3x^3y^2\sqrt{2y}.
\]
Thus, the expression in its simplest radical form is:
\[
\boxed{3x^3y^2\sqrt{2y}}.
\]
1. **Simplify the numerical part:**
- The prime factorization of 18 is \(2 \times 3^2\).
- Therefore, \(\sqrt{18} = \sqrt{2 \times 3^2} = 3\sqrt{2}\).
2. **Simplify the variable part:**
- For \(x^6\), we can extract squares: \(\sqrt{x^6} = x^{6/2} = x^3\).
- For \(y^5\), we can extract squares as well: \(\sqrt{y^5} = \sqrt{y^4 \times y} = \sqrt{y^4} \cdot \sqrt{y} = y^{4/2} \cdot \sqrt{y} = y^2\sqrt{y}\).
3. **Combine all parts:**
- Putting it all together, we get:
\[
\sqrt{18x^6y^5} = \sqrt{18} \cdot \sqrt{x^6} \cdot \sqrt{y^5} = 3\sqrt{2} \cdot x^3 \cdot (y^2\sqrt{y}) = 3x^3y^2\sqrt{2y}.
\]
Thus, the expression in its simplest radical form is:
\[
\boxed{3x^3y^2\sqrt{2y}}.
\]