▪ 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

[latex]\begin{array}{ccc} \sqrt{3x+18}=x & \\ \sqrt{x+3}=x-3 & \\ \sqrt{x+5}-\sqrt{x - 3}=2\end{array}[/latex]

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

To find extraneous solutions, find the restrictions using (radicand) [latex]\geq 0[/latex] for even index radical equations. There is no restriction for odd index radical equations.
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 resulting equation.
If a radical term still remains, repeat steps 1–2.
Check solutions if they are in the restriction or by substituting them into the original equation.

recall multiplying polynomial expressions

When squaring (or raising to any power) both sides of an equation as in step (2) above, don’t forget to apply the properties of exponents carefully and distribute all the terms appropriately.

[latex]\left(x + 3\right)^2 \neq x^2+9[/latex]

[latex]\left(x + 3\right)^2 = \left(x+3\right)\left(x+3\right)=x^2+6x+9[/latex]

The special form for perfect square trinomials comes in handy when solving radical equations.

[latex]\left(a + b\right)^2 = a^2 + 2ab + b^2[/latex]

[latex]\left(a - b\right)^2 = a^2 - 2ab + b^2[/latex]

This enables us to square binomials containing radicals by following the form.

[latex]\begin{align} \left(x - \sqrt{3x - 7}\right)^2 &= x^2 - 2\sqrt{3x-7}+\left(\sqrt{3x-7}\right)^2 \\ &=x^2 - 2\sqrt{3x-7}+3x-7\end{align}[/latex]

Example: Solving an Equation with One Radical

Solve each radical equation.

(1) [latex]\sqrt{x-3}=4[/latex]

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(2) [latex]\sqrt{2x+5}=-8[/latex]

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(3) [latex]\sqrt[3]{7x-1}+2=0[/latex]

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Try It

Solve each radical equation.

Example: Solving an Equation with One Radical

Solve [latex]\sqrt{15 - 2x}=x[/latex].

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Try It

Solve the radical equation: [latex]\sqrt{x+3}=3x - 1[/latex]

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[Optional] Example: Solving a Radical Equation Containing Two Radicals

Solve [latex]\sqrt{2x+3}+\sqrt{x - 2}=4[/latex].

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As this equation contains two radicals, we isolate one radical, eliminate it, and then isolate the second radical.

[latex]\begin{array}{llllll}\sqrt{2x+3}+\sqrt{x - 2}=4\hfill & \hfill & \\ \sqrt{2x+3}=4-\sqrt{x - 2}\hfill & \text{Subtract }\sqrt{x - 2}\text{ from both sides}.\hfill & \\ {\left(\sqrt{2x+3}\right)}^{2}={\left(4-\sqrt{x - 2}\right)}^{2}\hfill & \text{Square both sides}.\hfill \end{array}[/latex]

Use the perfect square formula to expand the right side: [latex]{\left(a-b\right)}^{2}={a}^{2}-2ab+{b}^{2}[/latex].

[latex]\begin{array}{lllllllllll}2x+3={\left(4\right)}^{2}-2\left(4\right)\sqrt{x - 2}+{\left(\sqrt{x - 2}\right)}^{2}\hfill & \hfill & \\ 2x+3=16 - 8\sqrt{x - 2}+\left(x - 2\right)\hfill & \hfill & \\ 2x+3=14+x - 8\sqrt{x - 2}\hfill & \text{Combine like terms}.\hfill & \\ x - 11=-8\sqrt{x - 2}\hfill & \text{Isolate the second radical}.\hfill & \\ {\left(x - 11\right)}^{2}={\left(-8\sqrt{x - 2}\right)}^{2}\hfill & \text{Square both sides}.\hfill & \\ {x}^{2}-22x+121=64\left(x - 2\right)\hfill & \hfill \end{array}[/latex]

Now that both radicals have been eliminated, set the quadratic equal to zero and solve.

[latex]\begin{array}{llllllllll}{x}^{2}-22x+121=64x - 128\hfill & \hfill & \\ {x}^{2}-86x+249=0\hfill & \hfill & \\ \left(x - 3\right)\left(x - 83\right)=0\hfill & \text{Factor and solve}.\hfill & \\ x=3\hfill & \hfill & \\ x=83\hfill & \hfill \end{array}[/latex]

The proposed solutions are [latex]x=3[/latex] and [latex]x=83[/latex]. Check each solution in the original equation.

[latex]\begin{array}{lllll}\sqrt{2x+3}+\sqrt{x - 2}=4\hfill & \\ \sqrt{2x+3}=4-\sqrt{x - 2}\hfill & \\ \sqrt{2\left(3\right)+3}=4-\sqrt{\left(3\right)-2}\hfill & \\ \sqrt{9}=4-\sqrt{1}\hfill \\ 3=3\hfill \end{array}[/latex]

One solution is [latex]x=3[/latex].

Check [latex]x=83[/latex].

[latex]\begin{array}{lllll}\sqrt{2x+3}+\sqrt{x - 2}=4\hfill & \\ \sqrt{2x+3}=4-\sqrt{x - 2}\hfill & \\ \sqrt{2\left(83\right)+3}=4-\sqrt{\left(83 - 2\right)}\hfill & \\ \sqrt{169}=4-\sqrt{81}\hfill & \\ 13\ne -5\hfill \end{array}[/latex]

The only solution is [latex]x=3[/latex]. We see that [latex]x=83[/latex] is an extraneous solution.

[Optional] Try It

Solve the equation with two radicals: [latex]\sqrt{3x+7}+\sqrt{x+2}=1[/latex].

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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] and [latex]{8}^{\frac{2}{3}}[/latex] is another way of writing [latex]\left(\sqrt[3]{8}\right)^2[/latex].

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].

recall rewriting expressions containing exponents

Recall the properties used to simplify expressions containing exponents. They work the same whether the exponent is an integer or a fraction.

It is helpful to remind yourself of these properties frequently throughout the course. They will by handy from now on in all the mathematics you’ll do.

Product Rule: [latex]{a}^{m}\cdot {a}^{n}={a}^{m+n}[/latex]

Quotient Rule: [latex]\dfrac{{a}^{m}}{{a}^{n}}={a}^{m-n}[/latex]

Power Rule: [latex]{\left({a}^{m}\right)}^{n}={a}^{m\cdot n}[/latex]

Zero Exponent: [latex]{a}^{0}=1[/latex]

Negative Exponent: [latex]{a}^{-n}=\dfrac{1}{{a}^{n}} \text{ and } {a}^{n}=\dfrac{1}{{a}^{-n}}[/latex]

Power of a Product: [latex]\left(ab\right)^n=a^nb^n[/latex]

Power of a Quotient: [latex]\left(\dfrac{a}{b}\right)^n=\dfrac{a^n}{b^n}[/latex]

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:

[latex]{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}[/latex]

Example: Evaluating a Number Raised to a Rational Exponent

Evaluate [latex]{8}^{\frac{2}{3}}[/latex].

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Evaluate [latex]{64}^{-\frac{1}{3}}[/latex].

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Example: Solving an Equation involving 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].

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Solve the equation [latex]{x}^{\frac{3}{2}}=125[/latex].

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Recall factoring when the gcf is a variable

Remember, when factoring a GCF (greatest common factor) from a polynomial expression, factor out the smallest power of the variable present in each term. This works whether the exponent on the variable is an integer or a fraction.

Example: Solving an Equation Involving Rational Exponents and Factoring

Solve [latex]3{x}^{\frac{3}{4}}={x}^{\frac{1}{2}}[/latex].

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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.

[latex]\begin{array}{ll}3{x}^{\frac{3}{4}}-\left({x}^{\frac{1}{2}}\right)={x}^{\frac{1}{2}}-\left({x}^{\frac{1}{2}}\right)\hfill & \\ 3{x}^{\frac{3}{4}}-{x}^{\frac{1}{2}}=0\hfill \end{array}[/latex]

Now, it looks like we should factor the left side, but what do we factor out? We can always factor the term with the lowest exponent. Rewrite [latex]{x}^{\frac{1}{2}}[/latex] as [latex]{x}^{\frac{2}{4}}[/latex]. Then, factor out [latex]{x}^{\frac{2}{4}}[/latex] from both terms on the left.

[latex]\begin{array}{ll}3{x}^{\frac{3}{4}}-{x}^{\frac{2}{4}}=0\hfill & \\ {x}^{\frac{2}{4}}\left(3{x}^{\frac{1}{4}}-1\right)=0\hfill \end{array}[/latex]

Where did [latex]{x}^{\frac{1}{4}}[/latex] come from? Remember, when we multiply two numbers with the same base, we add the exponents. Therefore, if we multiply [latex]{x}^{\frac{2}{4}}[/latex] back in using the distributive property, we get the expression we had before the factoring, which is what should happen. We need an exponent such that when added to [latex]\frac{2}{4}[/latex] equals [latex]\frac{3}{4}[/latex]. Thus, the exponent on x in the parentheses is [latex]\frac{1}{4}[/latex].

Let us continue. Now we have two factors and can use the zero factor theorem.

[latex]\begin{array}{llllllllllllll}{x}^{\frac{2}{4}}\left(3{x}^{\frac{1}{4}}-1\right)=0\hfill & \hfill & \\ {x}^{\frac{2}{4}}=0\hfill & \hfill & \\ x=0\hfill & \hfill & \\ 3{x}^{\frac{1}{4}}-1=0\hfill & \hfill & \\ 3{x}^{\frac{1}{4}}=1\hfill & \hfill & \\ {x}^{\frac{1}{4}}=\frac{1}{3}\hfill & \text{Divide both sides by 3}.\hfill & \\ {\left({x}^{\frac{1}{4}}\right)}^{4}={\left(\frac{1}{3}\right)}^{4}\hfill & \text{Raise both sides to the reciprocal of }\frac{1}{4}.\hfill & \\ x=\frac{1}{81}\hfill & \hfill \end{array}[/latex]

The two solutions are [latex]x=0[/latex], [latex]x=\frac{1}{81}[/latex].

Try It

Solve: [latex]{\left(x+5\right)}^{\frac{3}{2}}=8[/latex].

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Module 1: Equations and Inequalities