Equations With Radicals and Rational Exponents
Learning Objectives
- Solve a radical equation, identify extraneous solution
- Solve an equation with ratinal exponents
A General Note: Radical Equations
An equation containing terms with a variable in the radicand is called a radical equation.How To: Given a radical equation, solve it.
- Isolate the radical expression on one side of the equal sign. Put all remaining terms on the other side.
- If the radical is a square root, then square both sides of the equation. If it is a cube root, then raise both sides of the equation to the third power. In other words, for an nth root radical, raise both sides to the nth power. Doing so eliminates the radical symbol.
- Solve the remaining equation.
- If a radical term still remains, repeat steps 1–2.
- Confirm solutions by substituting them into the original equation.
Example: Solving an Equation with One Radical
Solve [latex]\sqrt{15 - 2x}=x[/latex].Answer: The radical is already isolated on the left side of the equal side, so proceed to square both sides.
Try It
Solve the radical equation: [latex]\sqrt{x+3}=3x - 1[/latex]Answer: [latex]x=1[/latex]; extraneous solution [latex]x=-\frac{2}{9}[/latex]
Solve Equations With 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.A General Note: 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:Example: Evaluating a Number Raised to a Rational Exponent
Evaluate [latex]{8}^{\frac{2}{3}}[/latex].Answer: 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].Answer: [latex]\frac{1}{4}[/latex]
Example: Solving an Equation Involving Rational Exponents and Factoring
Solve [latex]3{x}^{\frac{3}{4}}={x}^{\frac{1}{2}}[/latex].Answer: This equation involves rational exponents as well as factoring rational exponents. Let us take this one step at a time. First, put the variable terms on one side of the equal sign and set the equation equal to zero.
Try It
Solve: [latex]{\left(x+5\right)}^{\frac{3}{2}}=8[/latex].Answer: [latex]\{-1\}[/latex]
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