Sure! Let's simplify the expression [tex]\( \sqrt[6]{x^2 y^6} \)[/tex] step-by-step.
### Step 1: Understand the Expression
We are given the expression [tex]\( \sqrt[6]{x^2 y^6} \)[/tex], which is a 6th root of a product of two terms [tex]\( x^2 \)[/tex] and [tex]\( y^6 \)[/tex].
### Step 2: Separate the Radicals
We can use the property of radicals that states [tex]\( \sqrt[n]{a \cdot b} = \sqrt[n]{a} \cdot \sqrt[n]{b} \)[/tex]. Applying this to our expression:
[tex]\[
\sqrt[6]{x^2 y^6} = \sqrt[6]{x^2} \cdot \sqrt[6]{y^6}
\][/tex]
### Step 3: Simplify [tex]\( \sqrt[6]{y^6} \)[/tex]
Next, we recognize that [tex]\( y^6 \)[/tex] under the 6th root simplifies directly because:
[tex]\[
\sqrt[6]{y^6} = (y^6)^{\frac{1}{6}} = y^{6 \cdot \frac{1}{6}} = y^1 = y
\][/tex]
So this part simplifies to [tex]\( y \)[/tex].
### Step 4: Address [tex]\( \sqrt[6]{x^2} \)[/tex]
For [tex]\( \sqrt[6]{x^2} \)[/tex], we note that this cannot be simplified further using the given radicals.
### Step 5: Combine the Results
Putting the simplified parts together:
[tex]\[
\sqrt[6]{x^2 y^6} = \sqrt[6]{x^2} \cdot y
\][/tex]
Or more neatly:
[tex]\[
\sqrt[6]{x^2 y^6} = y \cdot \sqrt[6]{x^2}
\][/tex]
### Final Simplified Expression
The simplified expression, using radicals, is:
[tex]\[
\boxed{y \sqrt[6]{x^2}}
\][/tex]
