Sure, let's simplify these radical expressions step-by-step.
### (a) [tex]\(\sqrt[7]{y^7}\)[/tex]
To simplify the radical expression [tex]\(\sqrt[7]{y^7}\)[/tex]:
1. Recognize that taking the 7th root of [tex]\(y^7\)[/tex] essentially means finding the value of [tex]\(y\)[/tex] that, when raised to the power of 7, gives [tex]\(y^7\)[/tex].
2. In other words, [tex]\((y^7)^{1/7}\)[/tex].
3. According to the properties of exponents [tex]\((a^m)^{n} = a^{m \cdot n}\)[/tex]:
[tex]\[
(y^7)^{1/7} = y^{7 \cdot \frac{1}{7}} = y^1 = y
\][/tex]
Therefore, the simplified form of [tex]\(\sqrt[7]{y^7}\)[/tex] is:
[tex]\[
y
\][/tex]
### (b) [tex]\(\sqrt[4]{(1+z)^4}\)[/tex]
To simplify the radical expression [tex]\(\sqrt[4]{(1+z)^4}\)[/tex]:
1. Recognize that taking the 4th root of [tex]\((1+z)^4\)[/tex] essentially means finding the value that, when raised to the power of 4, gives [tex]\((1+z)^4\)[/tex].
2. In other words, [tex]\(((1+z)^4)^{1/4}\)[/tex].
3. According to the properties of exponents [tex]\((a^m)^{n} = a^{m \cdot n}\)[/tex]:
[tex]\[
((1+z)^4)^{1/4} = (1+z)^{4 \cdot \frac{1}{4}} = (1+z)^1 = 1+z
\][/tex]
Therefore, the simplified form of [tex]\(\sqrt[4]{(1+z)^4}\)[/tex] is:
[tex]\[
1 + z
\][/tex]
### Final Answer:
[tex]\[
\begin{aligned}
(a) \quad \sqrt[7]{y^7} &= y, \\
(b) \quad \sqrt[4]{(1+z)^4} &= 1+z.
\end{aligned}
\][/tex]
These are the simplified forms of the given radical expressions.
