91Ó°ÊÓ

In the following exercises, simplify. \(\sqrt[3]{81}-\sqrt[3]{192}\)

In the following exercises, simplify. \(\sqrt[3]{64 a^{10}}-\sqrt[3]{-216 a^{12}}\)

In the following exercises, simplify. \(\sqrt[4]{486 u^{7}}+\sqrt[4]{768 u^{3}}\)

Population growth The expression \(10 \cdot x^{n}\) models the growth of a mold population after \(n\) generations. There were 10 spores at the start, and each had \(x\) offspring. So \(10 \cdot x^{n}\) is the number of offspring at the fifth generation. At the fifth generation there were 10,240 offspring. Simplify the expression \(\sqrt[5]{\frac{10,240}{10}}\) to determine the number of offspring of each spore.

Spread of a virus The expression \(3 \cdot x^{n}\) models the spread of a virus after \(n\) cycles. There were three people originally infected with the virus, and each of them infected \(x\) people. So \(3 \cdot x^{4}\) is the number of people infected on the fourth cycle. At the fourth cycle 1875 people were infected. Simplify the expression \(\sqrt[4]{\frac{1875}{3}}\) to determine the number of people each person infected.

Explain why \(\sqrt[4]{-64}\) is not a real number but \(\sqrt[3]{-64}\) is.

In the following exercises, write as a radical expression. (a) \(x^{\frac{1}{2}}\) (b) \(y^{\frac{1}{3}}\) (c) \(z^{\frac{1}{4}}\)

In the following exercises, write as a radical expression. (a) \(r^{\frac{1}{2}}\) (b) \(s^{\frac{1}{3}}\) (c) \(t^{\frac{1}{4}}\)

In the following exercises, write as a radical expression. (a) \(u^{\frac{1}{5}}\) (b) \(v^{\frac{1}{9}}\) (c) \(w^{\frac{1}{20}}\)

In the following exercises, write as a radical expression. (a) \(g^{\frac{1}{7}}\) (b) \(h^{\frac{1}{5}}\) (c) \(j^{\frac{1}{25}}\)