2.6 Other types of equations

Page 1 / 10

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, $16^{\frac{1}{2}}$ is another way of writing $\sqrt{16};$ $8^{\frac{1}{3}}$ is another way of writing $ \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, $\frac{2}{3} (\frac{3}{2}) = 1,$ $3 (\frac{1}{3}) = 1,$ 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}} = {(a^{\frac{1}{n}})}^{m} = {(a^{m})}^{\frac{1}{n}} = \sqrt[n]{a^{m}} = {(\sqrt[n]{a})}^{m}

Evaluating a number raised to a rational exponent

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

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 $8^{\frac{2}{3}}$ as ${(8^{\frac{1}{3}})}^{2} .$

\begin{matrix} {(8^{\frac{1}{3}})}^{2} & = & {(2)}^{2} \\ = & 4 \end{matrix}

Got questions? Get instant answers now!

Try It

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

$\frac{1}{4}$

Got questions? Get instant answers now!

Solve the equation including a variable raised to a rational exponent

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

The way to remove the exponent on x is by raising both sides of the equation to a power that is the reciprocal of $\frac{5}{4},$ which is $\frac{4}{5} .$

\begin{matrix} x^{\frac{5}{4}} & = & 32 \\ {(x^{\frac{5}{4}})}^{\frac{4}{5}} & = & {(32)}^{\frac{4}{5}} \\ x & = & {(2)}^{4} & The fifth root of 32 is 2. \\ = & 16 \end{matrix}

Got questions? Get instant answers now!

Try It

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

$25$

Got questions? Get instant answers now!

Solving an equation involving rational exponents and factoring

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

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.

\begin{matrix} 3 x^{\frac{3}{4}} - (x^{\frac{1}{2}}) & = & x^{\frac{1}{2}} - (x^{\frac{1}{2}}) \\ 3 x^{\frac{3}{4}} - x^{\frac{1}{2}} & = & 0 \end{matrix}

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 $x^{\frac{1}{2}}$ as $x^{\frac{2}{4}} .$ Then, factor out $x^{\frac{2}{4}}$ from both terms on the left.

\begin{matrix} 3 x^{\frac{3}{4}} - x^{\frac{2}{4}} & = & 0 \\ x^{\frac{2}{4}} (3 x^{\frac{1}{4}} - 1) & = & 0 \end{matrix}

Where did $x^{\frac{1}{4}}$ come from? Remember, when we multiply two numbers with the same base, we add the exponents. Therefore, if we multiply $x^{\frac{2}{4}}$ 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 $\frac{2}{4}$ equals $\frac{3}{4} .$ Thus, the exponent on x in the parentheses is $\frac{1}{4} .$

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

\begin{matrix} x^{\frac{2}{4}} (3 x^{\frac{1}{4}} - 1) & = & 0 \\ x^{\frac{2}{4}} & = & 0 \\ x & = & 0 \\ 3 x^{\frac{1}{4}} - 1 & = & 0 \\ 3 x^{\frac{1}{4}} & = & 1 \\ x^{\frac{1}{4}} & = & \frac{1}{3} & Divide both sides by 3 . \\ {(x^{\frac{1}{4}})}^{4} & = & {(\frac{1}{3})}^{4} & Raise both sides to the reciprocal of \frac{1}{4} . \\ x & = & \frac{1}{81} \end{matrix}

The two solutions are $0$ and $\frac{1}{81} .$

Got questions? Get instant answers now!

Questions & Answers

Why is b in the answer

Dahsolar Reply

how do you work it out?

Brad Reply

answer

Ernest

heheheehe

Nitin

(Pcos∅+qsin∅)/(pcos∅-psin∅)

John Reply

how to do that?

Rosemary Reply

what is it about?

Amoah

how to answer the activity

Chabelita Reply

how to solve the activity

Chabelita

solve for X,,4^X-6(2^)-16=0

Alieu Reply

x4xminus 2

Lominate

sobhan Singh jina uniwarcity tignomatry ka long answers tile questions

harish Reply

t he silly nut company makes two mixtures of nuts: mixture a and mixture b. a pound of mixture a contains 12 oz of peanuts, 3 oz of almonds and 1 oz of cashews and sells for $4. a pound of mixture b contains 12 oz of peanuts, 2 oz of almonds and 2 oz of cashews and sells for $5. the company has 1080

ZAHRO Reply

If  , , are the roots of the equation 3 2 0, x px qx r     Find the value of 1  .

Swetha Reply

Parts of a pole were painted red, blue and yellow. 3/5 of the pole was red and 7/8 was painted blue. What part was painted yellow?

Patrick Reply

Parts of the pole was painted red, blue and yellow. 3 /5 of the pole was red and 7 /8 was painted blue. What part was painted yellow?

Patrick

how I can simplify algebraic expressions

Katleho Reply

Lairene and Mae are joking that their combined ages equal Sam’s age. If Lairene is twice Mae’s age and Sam is 69 yrs old, what are Lairene’s and Mae’s ages?

Mary Reply

23yrs

Yeboah

lairenea's age is 23yrs

ACKA

Katleho

Ello everyone

Katleho

Laurene is 46 yrs and Mae is 23 is

Solomon

hey people

christopher

age does not matter

christopher

solve for X, 4^x-6(2*)-16=0

Alieu

prove`x^3-3x-2cosA=0 (-π<A<=π

Mayank Reply

create a lesson plan about this lesson

Rose Reply

Excusme but what are you wrot?

<< Chapter < Page Page > Chapter >>

Practice Key Terms 5

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

100% Free Mobile Applications
Receive real-time job alerts and never miss the right job again

Source: OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6

Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask

	2 Other types of equations By OpenStax Read Online Course
	9 Sociology 09 Social Stratification in the US MCQ By OpenStax Start Quiz
	1 Biology 01 The Study of Life MCQ By OpenStax Start Quiz
	24 PLANT QUIZ 1 By Brooke Delaney Start Exam
©flickr: U.S.	Molecular Cellular Biology By Ann Schlosser Start Quiz
	Engineering Communication MCQ By Steve Gibbs Start Quiz
	31 Biology 31 Soil and Plant Nutrition MCQ By OpenStax Start Quiz
	Anthropology Aesthetics Culture By Richley Crapo Start Assignment
	Cardiac Electrophysiology Basic By Mistry Bhavesh Start Quiz
	Thermal-Fluid Systems MCQ By Steve Gibbs Start Quiz
	9 Psychology MCQ 2011 2 Exam By John Gabrieli Start Exam