RadF - Properties of Rational Exponents Lesson

Properties of Rational Exponents

The properties of integer exponents that you have previously learned can also be applied to rational exponents.

Properties of Rational Exponents

Let a and b be real numbers and let m and n be rational numbers.

The properties of rational exponents have the same application as the properties of integer exponents. Look at the following:

chart explaining properties of rational exponents

Simplest Form

Just as in all expressions you have studied in mathematics to this point, you will be expected to write radical expressions (whether in exponential form or in radical form) in simplest form. A radical with index n is in the simplest form if the radicand has no perfect nth powers as factors and any denominator has been rationalized.

Example: Use Properties of Exponents

Use the properties of rational exponents to simplify the expression.

a. equation image indicator $LaTeX: 3\frac{1}{2}\cdot3\frac{3}{2}=3^{\left(\frac{1}{2}+\frac{3}{2}\right)}=3^2=9$ $3\frac{1}{2}\cdot3\frac{3}{2}=3^{\left(\frac{1}{2}+\frac{3}{2}\right)}=3^2=9$

b. equation image indicator $LaTeX: \left(4^{\frac{4}{3}}\right)^3=4^{\left(\frac{4}{3}\cdot3\right)}=4^4=256$ $\left(4^{\frac{4}{3}}\right)^3=4^{\left(\frac{4}{3}\cdot3\right)}=4^4=256$

c. equation image indicator $LaTeX: \left(16\cdot100\right)^{\frac{1}{2}}=16^{\frac{1}{2}}\cdot100^{\frac{1}{2}}=4\cdot10=40$ $\left(16\cdot100\right)^{\frac{1}{2}}=16^{\frac{1}{2}}\cdot100^{\frac{1}{2}}=4\cdot10=40$

d. equation image indicator $LaTeX: \left(64\right)^{-\frac{1}{3}}=\frac{1}{\left(64\right)^{\frac{1}{3}}}=\frac{1}{4}$ $\left(64\right)^{-\frac{1}{3}}=\frac{1}{\left(64\right)^{\frac{1}{3}}}=\frac{1}{4}$

e. equation image indicator $LaTeX: \frac{5^{\frac{5}{4}}}{5^{\frac{1}{4}}}=5^{\left(\frac{5}{4}-\frac{1}{4}\right)}=5^{\frac{4}{4}}=5$ $\frac{5^{\frac{5}{4}}}{5^{\frac{1}{4}}}=5^{\left(\frac{5}{4}-\frac{1}{4}\right)}=5^{\frac{4}{4}}=5$

f. equation image indicator $LaTeX: \left(\frac{16}{625}\right)^{\frac{1}{4}}=\frac{16^{\frac{1}{4}}}{625^{\frac{1}{4}}}=\frac{2}{5}$ $\left(\frac{16}{625}\right)^{\frac{1}{4}}=\frac{16^{\frac{1}{4}}}{625^{\frac{1}{4}}}=\frac{2}{5}$

Example: Write Radicals in the Simplest Form

a. equation image indicator $LaTeX: \sqrt[4]{4,096}=\sqrt[4]{\left(8\right)^4}=8$ $\sqrt[4]{4,096}=\sqrt[4]{\left(8\right)^4}=8$

b. equation image indicator $LaTeX: \sqrt[4]{\frac{81}{16}}=\frac{\sqrt[4]{81}}{\sqrt[4]{16}}=\frac{3}{2}$ $\sqrt[4]{\frac{81}{16}}=\frac{\sqrt[4]{81}}{\sqrt[4]{16}}=\frac{3}{2}$

Properties of Rational Exponents Practice

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