## Equations With Radicals and Rational Exponents

### Learning Outcomes

• Solve a radical equation, identify extraneous solution.
• Solve an equation with rational exponents.

Radical equations are equations that contain variables in the radicand (the expression under a radical symbol), such as

$\begin{array}{ccc} \sqrt{3x+18}=x & \\ \sqrt{x+3}=x-3 & \\ \sqrt{x+5}-\sqrt{x - 3}=2\end{array}$

Radical equations may have one or more radical terms and are solved by eliminating each radical, one at a time. We have to be careful when solving radical equations as it is not unusual to find extraneous solutions, roots that are not, in fact, solutions to the equation. These solutions are not due to a mistake in the solving method, but result from the process of raising both sides of an equation to a power. Checking each answer in the original equation will confirm the true solutions.

### 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

1. Isolate the radical expression on one side of the equal sign. Put all remaining terms on the other side.
2. 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.
3. Solve the resulting equation.
4. If a radical term still remains, repeat steps 1–2.
5. Check solutions by substituting them into the original equation.

### Example: Solving an Equation with One Radical

Solve $\sqrt{15 - 2x}=x$.

### Try It

Solve the radical equation: $\sqrt{x+3}=3x - 1$

Solve $\sqrt{2x+3}+\sqrt{x - 2}=4$.

### Try It

Solve the equation with two radicals: $\sqrt{3x+7}+\sqrt{x+2}=1$.

## 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, ${16}^{\frac{1}{2}}$ is another way of writing $\sqrt{16}$; ${8}^{\frac{1}{3}}$ is another way of writing $\text{ }\sqrt[3]{8}$. 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, $\frac{2}{3}\left(\frac{3}{2}\right)=1$, $3\left(\frac{1}{3}\right)=1$, 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:

${a}^{\frac{m}{n}}={\left({a}^{\frac{1}{n}}\right)}^{m}={\left({a}^{m}\right)}^{\frac{1}{n}}=\sqrt[n]{{a}^{m}}={\left(\sqrt[n]{a}\right)}^{m}$

### Example: Evaluating a Number Raised to a Rational Exponent

Evaluate ${8}^{\frac{2}{3}}$.

### Try It

Evaluate ${64}^{-\frac{1}{3}}$.

### Example: Solve the Equation Including a Variable Raised to a Rational Exponent

Solve the equation in which a variable is raised to a rational exponent: ${x}^{\frac{5}{4}}=32$.

### Try It

Solve the equation ${x}^{\frac{3}{2}}=125$.

### Example: Solving an Equation Involving Rational Exponents and Factoring

Solve $3{x}^{\frac{3}{4}}={x}^{\frac{1}{2}}$.

### Try It

Solve: ${\left(x+5\right)}^{\frac{3}{2}}=8$.

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