To simplify the expression [tex]\(\sqrt[3]{(x + y)^4}\)[/tex], follow these steps:
1. Recognize the exponent under the cube root:
The expression under the radical is [tex]\((x + y)^4\)[/tex].
2. Interpret the cube root:
The cube root of a number or expression is equivalent to raising that number or expression to the power of [tex]\(\frac{1}{3}\)[/tex].
Therefore, we can rewrite [tex]\(\sqrt[3]{(x + y)^4}\)[/tex] as:
[tex]\[
\sqrt[3]{(x + y)^4} = ((x + y)^4)^{\frac{1}{3}}
\][/tex]
3. Combine the exponents:
Using the property of exponents that [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]:
[tex]\[
((x + y)^4)^{\frac{1}{3}} = (x + y)^{4 \cdot \frac{1}{3}}
\][/tex]
4. Simplify the exponent:
Multiply the exponents together:
[tex]\[
(x + y)^{4 \cdot \frac{1}{3}} = (x + y)^{\frac{4}{3}}
\][/tex]
Therefore, the simplified form of the given expression is:
[tex]\[
\boxed{(x + y)^{\frac{4}{3}}}
\][/tex]
