14 Other Types of Equations
Learning Objectives
In this section you will:
- Solve equations involving rational exponents.
- Solve equations using factoring.
- Solve radical equations.
- Solve absolute value equations.
- Solve other types of equations.
We have solved linear equations, rational equations, and quadratic equations using several methods. However, there are many other types of equations, and we will investigate a few more types in this section. We will look at equations involving rational exponents, polynomial equations, radical equations, absolute value equations, equations in quadratic form, and some rational equations that can be transformed into quadratics. Solving any equation, however, employs the same basic algebraic rules. We will learn some new techniques as they apply to certain equations, but the algebra never changes.
Solving Equations Involving Rational Exponents
Rational exponents are exponents that are fractions, where the numerator is a power and the denominator is a root. For example,[latex]\,{16}^{\frac{1}{2}}\,[/latex]is another way of writing[latex]\,\sqrt{16};[/latex][latex]{8}^{\frac{1}{3}}\,[/latex]is another way of writing[latex]\text{}\,\sqrt[3]{8}.\,[/latex]The ability to work with rational exponents is a useful skill, as it is highly applicable in calculus.
We can solve equations in which a variable is raised to a rational exponent by raising both sides of the equation to the reciprocal of the exponent. The reason we raise the equation to the reciprocal of the exponent is because we want to eliminate the exponent on the variable term, and a number multiplied by its reciprocal equals 1. For example,[latex]\,\frac{2}{3}\left(\frac{3}{2}\right)=1,[/latex][latex]3\left(\frac{1}{3}\right)=1,[/latex]and so on.
Rational Exponents
A rational exponent indicates a power in the numerator and a root in the denominator. There are multiple ways of writing an expression, a variable, or a number with a rational exponent:
Evaluating a Number Raised to a Rational Exponent
Evaluate[latex]\,{8}^{\frac{2}{3}}.[/latex]
[hidden-answer a=”fs-id2422284″]
Whether we take the root first or the power first depends on the number. It is easy to find the cube root of 8, so rewrite[latex]\,{8}^{\frac{2}{3}}\,[/latex]as[latex]\,{\left({8}^{\frac{1}{3}}\right)}^{2}.[/latex]
Try It
Evaluate[latex]\,{64}^{-\frac{1}{3}}.[/latex]
[hidden-answer a=”fs-id2434546″]
[latex]\frac{1}{4}[/latex]
[/hidden-answer]
Solve the Equation Including a Variable Raised to a Rational Exponent
Solve the equation in which a variable is raised to a rational exponent:[latex]\,{x}^{\frac{5}{4}}=32.[/latex]
[hidden-answer a=”fs-id1569151″]
The way to remove the exponent on x is by raising both sides of the equation to a power that is the reciprocal of[latex]\,\frac{5}{4},[/latex]which is[latex]\,\frac{4}{5}.[/latex]
Try It
Solve the equation[latex]\,{x}^{\frac{3}{2}}=125.[/latex]
[latex]25[/latex][/hidden-answer]
Solving an Equation Involving Rational Exponents and Facto