# 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, $\text{\hspace{0.17em}}{16}^{\frac{1}{2}}\text{\hspace{0.17em}}$ is another way of writing $\text{\hspace{0.17em}}\sqrt{16};$ ${8}^{\frac{1}{3}}\text{\hspace{0.17em}}$ is another way of writing $\text{​}\text{\hspace{0.17em}}\sqrt[3]{8}.\text{\hspace{0.17em}}$ 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, $\text{\hspace{0.17em}}\frac{2}{3}\left(\frac{3}{2}\right)=1,$ $3\left(\frac{1}{3}\right)=1,$ 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:

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

## Evaluating a number raised to a rational exponent

Evaluate $\text{\hspace{0.17em}}{8}^{\frac{2}{3}}.$

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 $\text{\hspace{0.17em}}{8}^{\frac{2}{3}}\text{\hspace{0.17em}}$ as $\text{\hspace{0.17em}}{\left({8}^{\frac{1}{3}}\right)}^{2}.$

$\begin{array}{ccc}\hfill {\left({8}^{\frac{1}{3}}\right)}^{2}& =\hfill & {\left(2\right)}^{2}\hfill \\ & =& 4\hfill \end{array}$

Evaluate $\text{\hspace{0.17em}}{64}^{-\frac{1}{3}}.$

$\frac{1}{4}$

## Solve the equation including a variable raised to a rational exponent

Solve the equation in which a variable is raised to a rational exponent: $\text{\hspace{0.17em}}{x}^{\frac{5}{4}}=32.$

The way to remove the exponent on x is by raising both sides of the equation to a power that is the reciprocal of $\text{\hspace{0.17em}}\frac{5}{4},$ which is $\text{\hspace{0.17em}}\frac{4}{5}.$

Solve the equation $\text{\hspace{0.17em}}{x}^{\frac{3}{2}}=125.$

$25$

## Solving an equation involving rational exponents and factoring

Solve $\text{\hspace{0.17em}}3{x}^{\frac{3}{4}}={x}^{\frac{1}{2}}.$

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.

$\begin{array}{ccc}\hfill 3{x}^{\frac{3}{4}}-\left({x}^{\frac{1}{2}}\right)& =& {x}^{\frac{1}{2}}-\left({x}^{\frac{1}{2}}\right)\hfill \\ \hfill 3{x}^{\frac{3}{4}}-{x}^{\frac{1}{2}}& =& 0\hfill \end{array}$

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 $\text{\hspace{0.17em}}{x}^{\frac{1}{2}}\text{\hspace{0.17em}}$ as $\text{\hspace{0.17em}}{x}^{\frac{2}{4}}.\text{\hspace{0.17em}}$ Then, factor out $\text{\hspace{0.17em}}{x}^{\frac{2}{4}}\text{\hspace{0.17em}}$ from both terms on the left.

$\begin{array}{ccc}\hfill 3{x}^{\frac{3}{4}}-{x}^{\frac{2}{4}}& =& 0\hfill \\ \hfill {x}^{\frac{2}{4}}\left(3{x}^{\frac{1}{4}}-1\right)& =& 0\hfill \end{array}$

Where did $\text{\hspace{0.17em}}{x}^{\frac{1}{4}}\text{\hspace{0.17em}}$ come from? Remember, when we multiply two numbers with the same base, we add the exponents. Therefore, if we multiply $\text{\hspace{0.17em}}{x}^{\frac{2}{4}}\text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}\frac{2}{4}\text{\hspace{0.17em}}$ equals $\text{\hspace{0.17em}}\frac{3}{4}.\text{\hspace{0.17em}}$ Thus, the exponent on x in the parentheses is $\text{\hspace{0.17em}}\frac{1}{4}.\text{\hspace{0.17em}}$

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

The two solutions are $\text{\hspace{0.17em}}0$ and $\frac{1}{81}.$

#### Questions & Answers

the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
Kc Reply
1+cos²A/cos²A=2cosec²A-1
Ramesh Reply
test for convergence the series 1+x/2+2!/9x3
success Reply
a man walks up 200 meters along a straight road whose inclination is 30 degree.How high above the starting level is he?
Lhorren Reply
100 meters
Kuldeep
Find that number sum and product of all the divisors of 360
jancy Reply
answer
Ajith
exponential series
Naveen
what is subgroup
Purshotam Reply
Prove that: (2cos&+1)(2cos&-1)(2cos2&-1)=2cos4&+1
Macmillan Reply
e power cos hyperbolic (x+iy)
Vinay Reply
10y
Michael
tan hyperbolic inverse (x+iy)=alpha +i bita
Payal Reply
prove that cos(π/6-a)*cos(π/3+b)-sin(π/6-a)*sin(π/3+b)=sin(a-b)
Tejas Reply
why {2kπ} union {kπ}={kπ}?
Huy Reply
why is {2kπ} union {kπ}={kπ}? when k belong to integer
Huy
if 9 sin theta + 40 cos theta = 41,prove that:41 cos theta = 41
Trilochan Reply
what is complex numbers
Ayushi Reply
Please you teach
Dua
Yes
ahmed
Thank you
Dua
give me treganamentry question
Anshuman Reply
Solve 2cos x + 3sin x = 0.5
shobana Reply

### Read also:

#### Get the best Algebra and trigonometry course in your pocket!

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?

 By By David Martin By Qqq Qqq