To simplify the expression [tex]\(\sqrt{x^2 y^6}\)[/tex], let's go through the following steps:
1. Understand the Expression Inside the Square Root:
The expression inside the square root is [tex]\(x^2 y^6\)[/tex].
2. Break Down the Terms:
We can break down the terms inside the square root based on their properties:
[tex]\[\sqrt{x^2 y^6} = \sqrt{x^2 \cdot y^6}\][/tex]
3. Apply the Square Root to Each Part:
The square root of a product is the product of the square roots:
[tex]\[\sqrt{x^2 \cdot y^6} = \sqrt{x^2} \cdot \sqrt{y^6}\][/tex]
4. Simplify Each Square Root Separately:
- For [tex]\(\sqrt{x^2}\)[/tex]:
[tex]\[\sqrt{x^2} = |x|\][/tex]
So, we have [tex]\(|x|\)[/tex] because the square root of [tex]\(x^2\)[/tex] is [tex]\(x\)[/tex] in absolute terms (it can be both positive or negative).
- For [tex]\(\sqrt{y^6}\)[/tex]:
[tex]\[\sqrt{y^6} = y^3\][/tex]
This follows because [tex]\((y^3)^2 = y^6\)[/tex], so the square root of [tex]\(y^6\)[/tex] is [tex]\(y^3\)[/tex].
5. Combine the Simplified Parts:
Putting these together, we get:
[tex]\[\sqrt{x^2 y^6} = |x| \cdot y^3\][/tex]
Hence, the simplified form of [tex]\(\sqrt{x^2 y^6}\)[/tex] is:
[tex]\[|x| y^3\][/tex]
