# 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

what are you up to?
nothing up todat yet
Miranda
hi
jai
hello
jai
Miranda Drice
jai
aap konsi country se ho
jai
which language is that
Miranda
I am living in india
jai
good
Miranda
what is the formula for calculating algebraic
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda
state and prove Cayley hamilton therom
hello
Propessor
hi
Miranda
the Cayley hamilton Theorem state if A is a square matrix and if f(x) is its characterics polynomial then f(x)=0 in another ways evey square matrix is a root of its chatacteristics polynomial.
Miranda
hi
jai
hi Miranda
jai
thanks
Propessor
welcome
jai
What is algebra
algebra is a branch of the mathematics to calculate expressions follow.
Miranda
Miranda Drice would you mind teaching me mathematics? I think you are really good at math. I'm not good at it. In fact I hate it. 😅😅😅
Jeffrey
lolll who told you I'm good at it
Miranda
something seems to wispher me to my ear that u are good at it. lol
Jeffrey
lolllll if you say so
Miranda
but seriously, Im really bad at math. And I hate it. But you see, I downloaded this app two months ago hoping to master it.
Jeffrey
which grade are you in though
Miranda
oh woww I understand
Miranda
haha. already finished college
Jeffrey
how about you? what grade are you now?
Jeffrey
I'm going to 11grade
Miranda
how come you finished in college and you don't like math though
Miranda
gotta practice, holmie
Steve
if you never use it you won't be able to appreciate it
Steve
I don't know why. But Im trying to like it.
Jeffrey
yes steve. you're right
Jeffrey
so you better
Miranda
what is the solution of the given equation?
which equation
Miranda
I dont know. lol
Jeffrey
please where is the equation
Miranda
Jeffrey
answer and questions in exercise 11.2 sums
how do u calculate inequality of irrational number?
Alaba
give me an example
Chris
and I will walk you through it
Chris
cos (-z)= cos z .
cos(- z)=cos z
Mustafa
what is a algebra
(x+x)3=?
6x
Obed
what is the identity of 1-cos²5x equal to?
__john __05
Kishu
Hi
Abdel
hi
Ye
hi
Nokwanda
C'est comment
Abdel
Hi
Amanda
hello
SORIE
Hiiii
Chinni
hello
Ranjay
hi
ANSHU
hiiii
Chinni
h r u friends
Chinni
yes
Hassan
so is their any Genius in mathematics here let chat guys and get to know each other's
SORIE
I speak French
Abdel
okay no problem since we gather here and get to know each other
SORIE
hi im stupid at math and just wanna join here
Yaona
lol nahhh none of us here are stupid it's just that we have Fast, Medium, and slow learner bro but we all going to work things out together
SORIE
it's 12
what is the function of sine with respect of cosine , graphically
tangent bruh
Steve
cosx.cos2x.cos4x.cos8x
sinx sin2x is linearly dependent
what is a reciprocal
The reciprocal of a number is 1 divided by a number. eg the reciprocal of 10 is 1/10 which is 0.1
Shemmy
Reciprocal is a pair of numbers that, when multiplied together, equal to 1. Example; the reciprocal of 3 is ⅓, because 3 multiplied by ⅓ is equal to 1
Jeza
each term in a sequence below is five times the previous term what is the eighth term in the sequence
I don't understand how radicals works pls
How look for the general solution of a trig function