In Watch (Roots and Exponents), radical expressions were written in simplified exponential form with \(\sqrt4=\sqrt{2^2}=2\) and \(\sqrt[3]{125}=\sqrt[3]{5^3}=5\). One method for simplifying roots is to prime factor the value under the radical and express as the powers of primes, then apply the root, and then rewrite the remaining root.
For example: \(\sqrt[3]{16}=\sqrt[3]{2\cdot 2\cdot 2\cdot 2}=\sqrt[3]{2^3\cdot 2} =2\sqrt[3]{2}\).
Similarly, this method can be used to write the root in simplified exponential form.
For example: \(\sqrt[3]{16}=\sqrt[3]{2\cdot 2\cdot 2\cdot 2}=\sqrt[3]{2^4} = 2^\frac{4}{3}\).
Rewrite each of the following roots into simplified root form and simplified exponential form.
1. \(\sqrt[4]4\)
2. \(\sqrt[3]8\)
3. \(\sqrt[3]{625}\)
4. \(\sqrt[4]{32}\)
Check your solutions here.
