To solve the expression [tex]\(\sqrt{x^6 y^2}\)[/tex], we will follow these steps:
1. Express the square root of a product:
The square root of a product is the product of the square roots. So, we can rewrite the expression as:
[tex]\[
\sqrt{x^6 y^2} = \sqrt{x^6} \cdot \sqrt{y^2}
\][/tex]
2. Simplify each square root individually:
- Square root of [tex]\(x^6\)[/tex]:
The square root of [tex]\(x^6\)[/tex] can be expressed by finding a power such that it is halved. Remember that [tex]\(\sqrt{x^a} = x^{a/2}\)[/tex]. Therefore:
[tex]\[
\sqrt{x^6} = x^{6/2} = x^3
\][/tex]
- Square root of [tex]\(y^2\)[/tex]:
Similarly, the square root of [tex]\(y^2\)[/tex] can be simplified as follows:
[tex]\[
\sqrt{y^2} = y^{2/2} = y^1 = y
\][/tex]
3. Combine the simplified results:
Now that we have simplified each of the square roots, we can multiply them together:
[tex]\[
\sqrt{x^6 y^2} = x^3 \cdot y
\][/tex]
Therefore, the simplified form of [tex]\(\sqrt{x^6 y^2}\)[/tex] is:
[tex]\[
\boxed{x^3 y}
\][/tex]
