RadF - Applying the Properties of Rational Exponents (Addition, Subtraction, and Multiplication) Lesson

Applying the Properties of Rational Exponents (Addition, Subtraction, and Multiplication)

Simplest Form

When you are applying the properties of rational exponents (radical expressions written in exponential form), you must also write your final answer in simplest form. Remember that a radical with index n is in simplest form if the radicand has no perfect nth powers as factors and any denominator has been rationalized.

We will now look at simplifying radical expressions, whether written in exponential form or radical form, which contain constants, variables, or a combination of both. We will also look at the addition, subtraction, multiplication, and division of radical expressions, as well as the application of radical expressions using a combination of any and or all operations.

Like Radicals

Radical expressions with the same index and radicand are like radicals. To add or subtract like radicals, use the distributive property. When radical expressions have variables within them, remember the following in simplifying:

Square Roots - Exponents must be multiples of "2".
Cube Roots - Exponents must be multiples of "3".
4th Roots - Exponents must be multiples of "4".

Example: Use Properties of Radicals

Use the properties of radicals to simplify the expression.

equation image indicator $LaTeX: \sqrt[3]{135m^7} \\ \sqrt[3]{27\cdot5}\cdot\sqrt[3]{m^6\cdot m^1} \\ \sqrt[3]{27}\cdot\sqrt[3]{5}\cdot\sqrt[3]{m^6}\cdot\sqrt[3]{m^1} \\ 3\sqrt[3]{5}\cdot m^2\cdot\sqrt[3]{m} \\ 3m^2\sqrt[3]{5m}$ $\sqrt[3]{135m^7} \\ \sqrt[3]{27\cdot5}\cdot\sqrt[3]{m^6\cdot m^1} \\ \sqrt[3]{27}\cdot\sqrt[3]{5}\cdot\sqrt[3]{m^6}\cdot\sqrt[3]{m^1} \\ 3\sqrt[3]{5}\cdot m^2\cdot\sqrt[3]{m} \\ 3m^2\sqrt[3]{5m}$

Example: Use Properties of Radicals

a. equation image indicator $LaTeX: \sqrt[4]{10}+7\sqrt[4]{10} \\ =\left(1+7\right)\sqrt[4]{10} \\ =8\sqrt[4]{10}$ $\sqrt[4]{10}+7\sqrt[4]{10} \\ =\left(1+7\right)\sqrt[4]{10} \\ =8\sqrt[4]{10}$

b. equation image indicator $LaTeX: 2\left(8^{\frac{1}{5}}\right)+10\left(8^{\frac{1}{5}}\right) \\ \left(2+10\right)\left(8^{\frac{1}{5}}\right) \\ =12\left(8^{\frac{1}{5}}\right)$ $2\left(8^{\frac{1}{5}}\right)+10\left(8^{\frac{1}{5}}\right) \\ \left(2+10\right)\left(8^{\frac{1}{5}}\right) \\ =12\left(8^{\frac{1}{5}}\right)$

c. equation image indicator $LaTeX: \sqrt[3]{54}-\sqrt[3]{2} \\ =\sqrt[3]{27}\cdot\sqrt[3]{2}-\sqrt[3]{2} \\ =3\sqrt[3]{2}-\sqrt[3]{2} \\ =\left(3-1\right)\sqrt[3]{2} \\ =2\sqrt[3]{2}$ $\sqrt[3]{54}-\sqrt[3]{2} \\ =\sqrt[3]{27}\cdot\sqrt[3]{2}-\sqrt[3]{2} \\ =3\sqrt[3]{2}-\sqrt[3]{2} \\ =\left(3-1\right)\sqrt[3]{2} \\ =2\sqrt[3]{2}$

Properties of Rational Exponents Applied to Radicals

Let a and b be real numbers and let m and n be rational numbers. The properties of radicals have the same application as the properties of integers of exponents.

Multiplying Radicals

When multiplying radicals remember the following steps:

Multiply the coefficients, then use the Product Property of Radicals, which is $LaTeX: \sqrt[n]{a\cdot b}=\sqrt[n]{a}\cdot\sqrt[n]{b}$ $\sqrt[n]{a\cdot b}=\sqrt[n]{a}\cdot\sqrt[n]{b}$
Simplify the resulting radical.

Example: Write Radicals in Simplest Form

$LaTeX: \sqrt[3]{4}\cdot\sqrt[3]{16}=\sqrt[3]{64}=4$ $\sqrt[3]{4}\cdot\sqrt[3]{16}=\sqrt[3]{64}=4$

equation image indicator $LaTeX: -3\sqrt[4]{64}\cdot-\sqrt[4]{8} \\ =\left(-3\cdot-1\right)\cdot\left(\sqrt[4]{4^3}\cdot\sqrt[4]{4^1\cdot2}\right) \\ =3\cdot\left(\sqrt[4]{4^4}\cdot\sqrt[4]{2}\right) \\ =3\cdot4\left(\sqrt[4]{2}\right) \\ =12\cdot\left(\sqrt[4]{2}\right)$ $-3\sqrt[4]{64}\cdot-\sqrt[4]{8} \\ =\left(-3\cdot-1\right)\cdot\left(\sqrt[4]{4^3}\cdot\sqrt[4]{4^1\cdot2}\right) \\ =3\cdot\left(\sqrt[4]{4^4}\cdot\sqrt[4]{2}\right) \\ =3\cdot4\left(\sqrt[4]{2}\right) \\ =12\cdot\left(\sqrt[4]{2}\right)$

equation image indicator $LaTeX: 5\sqrt[4]{a^4\cdot b^3}\cdot7\sqrt[4]{a^9\cdot b^7} \\ =\left(5\cdot7\right)\cdot\left(\sqrt[4]{a^4\cdot a^9}\cdot\sqrt[4]{b^3\cdot b^7}\right) \\ =35\left(\sqrt[4]{a^{4+9}}\cdot\sqrt[4]{b^{3+7}}\right) \\ =35\left(\sqrt[4]{a^{12}}\cdot\sqrt[4]{a^1}\right)\cdot\left(\sqrt[4]{b^8}\cdot\sqrt[4]{b^2}\right) \\ =35|a^3\cdot b^2|\cdot\left(\sqrt[4]{a}\cdot\sqrt[4]{b^2}\right) \\ =35|a^3\cdot b^2|\cdot\left(\sqrt[4]{a}\cdot\sqrt[]{b}\right)$ $5\sqrt[4]{a^4\cdot b^3}\cdot7\sqrt[4]{a^9\cdot b^7} \\ =\left(5\cdot7\right)\cdot\left(\sqrt[4]{a^4\cdot a^9}\cdot\sqrt[4]{b^3\cdot b^7}\right) \\ =35\left(\sqrt[4]{a^{4+9}}\cdot\sqrt[4]{b^{3+7}}\right) \\ =35\left(\sqrt[4]{a^{12}}\cdot\sqrt[4]{a^1}\right)\cdot\left(\sqrt[4]{b^8}\cdot\sqrt[4]{b^2}\right) \\ =35|a^3\cdot b^2|\cdot\left(\sqrt[4]{a}\cdot\sqrt[4]{b^2}\right) \\ =35|a^3\cdot b^2|\cdot\left(\sqrt[4]{a}\cdot\sqrt[]{b}\right)$

Example: Apply Properties of Exponents

A mammal's surface area S (in square cm) can be approximated by the model: equation image indicator $LaTeX: S=k\cdot m^{\frac{2}{3}}$ $S=k\cdot m^{\frac{2}{3}}$ where m is the mass (in grams) of the mammal and k is the constant. The values of k for some mammals are shown below. Approximate the surface area of a rabbit that has a mass of 3.4 kilograms $LaTeX: \left(3.4\times10^3grams\right)$ $\left(3.4\times10^3grams\right)$ .

Mammal	Sheep	Rabbit	Horse	Human	Monkey	Bat
k	8.4	9.75	10.0	11.0	11.8	57.5

Solution

equation image indicator $LaTeX: S=k\cdot m^{\frac{2}{3}} \\ S=9.75\cdot\left(3.4\times10^3\right)^{\frac{2}{3}} \\ S=9.75\cdot3.4^{\frac{2}{3}}\cdot\left(10^3\right)^{\frac{2}{3}} \\ S\approx9.75\cdot2.6\cdot10^2\approx2200$ $S=k\cdot m^{\frac{2}{3}} \\ S=9.75\cdot\left(3.4\times10^3\right)^{\frac{2}{3}} \\ S=9.75\cdot3.4^{\frac{2}{3}}\cdot\left(10^3\right)^{\frac{2}{3}} \\ S\approx9.75\cdot2.6\cdot10^2\approx2200$

Therefore, the rabbit's surface area is approximately 2200 square cm.

Add, Subtract, & Multiply Radical Expressions Practice

*For the practice problems #5, the correct solution is 15 |b^3| a^4 "times the fourth root of ab"*

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