2.7 Rational Exponents

Leanne Thompson

2.7 Rational Exponents

All powers with a rational exponent can be written in radical form.

Table 2.7.1
To write a power with a rational exponent in radical form:
$x^{\frac{b}{a}}=\sqrt[a]{x^b}\text{ or }(\sqrt[a]{x})^b$ The base of the power becomes the radicand. The numerator of the exponent in the power becomes the index. The denominator of the exponent in the power becomes the exponent of the radicand or the exponent of the entire radical.

Below are some examples of powers with rational exponents being written in radical form:

$x^{\frac{3}{2}}=\sqrt{x^3}\hspace{0.25in} x^{\frac{2}{3}}=\sqrt[3]{x^2}\hspace{0.25in} x^{\frac{5}{4}}=\sqrt[4]{x^5}\hspace{0.25in} x^{\frac{9}{5}}=\sqrt[5]{x^9}$

Practice 1

Write the following in radical form.

a) ${x}^{\frac{1}{2}}$

b) ${x}^{\frac{1}{3}}$

c) ${x}^{\frac{1}{4}}$

d) ${x}^{\frac{1}{5}}$

Table 2.7.2
To write a power with a negative rational exponent in radical form:
$x^{-\frac{b}{a}}=\dfrac{1}{\sqrt[a]{x^b}}\text{ or }\dfrac{1}{(\sqrt[a]{x})^b}$

Practice 2

Write the following in radical form, then evaluate.

a) ${8}^{\frac{2}{3}}$

b) ${16}^{\frac{1}{4}}$

c) ${25}^{-\frac{3}{2}}$

d) ${27}^{-\frac{4}{3}}$

Practice 3

Write the following in radical form, then evaluate.

a) ${(-64)}^{\frac{1}{3}}$

b) ${-64}^{\frac{1}{3}}$

c) ${(64)}^{-\frac{1}{3}}$

Practice 4

Evaluate the following using a calculator. Round to the nearest hundredth.

a) ${5}^{\frac{2}{3}}$

b) ${7}^{\frac{1}{4}}$

c) ${(3)}^{-\frac{5}{6}}$

d) ${(2)}^{-\frac{3}{2}}$

Practice 5

Use exponent laws to simplify. Write your answers with positive exponents.

a) $(y^{\frac{2}{3}})(y^\frac{4}{3})$

b) $(z^{\frac{7}{4}})(z^\frac{5}{4})$

Practice 6

Use exponent laws to simplify. Write your answers with positive exponents.

a) $(a^{\frac{3}{4}}b^{\frac{1}{4}})^{-2}$

b) $(x^{-\frac{2}{5}}y^{\frac{5}{3}})^{-1}$

Homework

Write the following powers in radical form.

a) $m^{\frac{3}{5}}$

b) $(10r)^{-\frac{3}{4}}$

c) $(7x)^{\frac{3}{2}}$

d) $(6b)^{-\frac{4}{3}}$
Write each of the following radicals as a power.

a) $\sqrt[3]{5}$

b) $\sqrt[5]{2^3}$

c) $\sqrt[3]{a}$

d) $\sqrt[5]{x^3}$
Evaluate the following. Round to the nearest hundredth where necessary.

a) $8^{\frac{2}{3}}$

b) $16^{\frac{1}{4}}$

c) $\sqrt[3]{4^6}$

d) $\sqrt[5]{32^2}$

e) $2^{-\frac{4}{5}}$

f) $6^{\frac{1}{3}}$

g) $-9^{\frac{2}{7}}$

h) $(-3)^{\frac{5}{3}}$
Use exponent laws to simplify. Write your answers with positive exponents.

a) $(xy^{\frac{1}{3}})(xy^{\frac{2}{3}})$

b) $(4v^{\frac{1}{4}})(v^{\frac{7}{4}})$

c) $(a^{\frac{1}{2}}b^{\frac{1}{2}})^{-1}$

d) $(x^{\frac{5}{3}}y^{-2})^0$