To solve the expression [tex]\(\sqrt[7]{4 x^2 y} \cdot \sqrt[7]{12 x^4 y^4}\)[/tex] using the product rule for radicals, we can follow these steps:
1. Combine the radicals under a common radical expression.
According to the product rule for radicals, [tex]\(\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a \cdot b}\)[/tex]. For our specific case:
[tex]\[
\sqrt[7]{4 x^2 y} \cdot \sqrt[7]{12 x^4 y^4} = \sqrt[7]{(4 x^2 y) \cdot (12 x^4 y^4)}
\][/tex]
2. Multiply the expressions inside the radical.
First, multiply the constants:
[tex]\[
4 \cdot 12 = 48
\][/tex]
Next, multiply the variables by adding their exponents, as per the rules of exponents when multiplying like bases.
For [tex]\(x\)[/tex]:
[tex]\[
x^2 \cdot x^4 = x^{2+4} = x^6
\][/tex]
For [tex]\(y\)[/tex]:
[tex]\[
y \cdot y^4 = y^{1+4} = y^5
\][/tex]
So, we combine everything together:
[tex]\[
(4 x^2 y) \cdot (12 x^4 y^4) = 48 x^6 y^5
\][/tex]
3. Write the result under a single radical.
Therefore, the combined expression under the seventh root is:
[tex]\[
\sqrt[7]{48 x^6 y^5}
\][/tex]
Putting it all together, the simplified expression is:
[tex]\[
\sqrt[7]{4 x^2 y} \cdot \sqrt[7]{12 x^4 y^4} = \sqrt[7]{48 x^6 y^5}
\][/tex]
This is the final answer.
