2.6 Other types of equations

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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}

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

$\frac{1}{4}$

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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}

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

$25$

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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 $\frac{1}{81} .$

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Questions & Answers

Three charges q_{1}=+3\mu C, q_{2}=+6\mu C and q_{3}=+8\mu C are located at (2,0)m (0,0)m and (0,3) coordinates respectively. Find the magnitude and direction acted upon q_{2} by the two other charges.Draw the correct graphical illustration of the problem above showing the direction of all forces.

Kate Reply

To solve this problem, we need to first find the net force acting on charge q_{2}. The magnitude of the force exerted by q_{1} on q_{2} is given by F=\frac{kq_{1}q_{2}}{r^{2}} where k is the Coulomb constant, q_{1} and q_{2} are the charges of the particles, and r is the distance between them.

Muhammed

What is the direction and net electric force on q_{1}= 5µC located at (0,4)r due to charges q_{2}=7mu located at (0,0)m and q_{3}=3\mu C located at (4,0)m?

Kate Reply

what is the change in momentum of a body?

Eunice Reply

what is a capacitor?

Raymond Reply

Capacitor is a separation of opposite charges using an insulator of very small dimension between them. Capacitor is used for allowing an AC (alternating current) to pass while a DC (direct current) is blocked.

Gautam

A motor travelling at 72km/m on sighting a stop sign applying the breaks such that under constant deaccelerate in the meters of 50 metres what is the magnitude of the accelerate

Maria Reply

please solve

Sharon

8m/s²

Aishat

What is Thermodynamics

Muordit

velocity can be 72 km/h in question. 72 km/h=20 m/s, v^2=2.a.x , 20^2=2.a.50, a=4 m/s^2.

Mehmet

A boat travels due east at a speed of 40meter per seconds across a river flowing due south at 30meter per seconds. what is the resultant speed of the boat

Saheed Reply

50 m/s due south east

Someone

which has a higher temperature, 1cup of boiling water or 1teapot of boiling water which can transfer more heat 1cup of boiling water or 1 teapot of boiling water explain your . answer

Ramon Reply

I believe temperature being an intensive property does not change for any amount of boiling water whereas heat being an extensive property changes with amount/size of the system.

Someone

Scratch that

Someone

temperature for any amount of water to boil at ntp is 100⁰C (it is a state function and and intensive property) and it depends both will give same amount of heat because the surface available for heat transfer is greater in case of the kettle as well as the heat stored in it but if you talk.....

Someone

about the amount of heat stored in the system then in that case since the mass of water in the kettle is greater so more energy is required to raise the temperature b/c more molecules of water are present in the kettle

Someone

definitely of physics

Haryormhidey Reply

how many start and codon

Esrael Reply

what is field

Felix Reply

physics, biology and chemistry this is my Field

ALIYU

field is a region of space under the influence of some physical properties

Collete

what is ogarnic chemistry

WISDOM Reply

determine the slope giving that 3y+ 2x-14=0

WISDOM

Another formula for Acceleration

Belty Reply

a=v/t. a=f/m a

IHUMA

innocent

Adah

pratica A on solution of hydro chloric acid,B is a solution containing 0.5000 mole ofsodium chlorid per dm³,put A in the burret and titrate 20.00 or 25.00cm³ portion of B using melting orange as the indicator. record the deside of your burret tabulate the burret reading and calculate the average volume of acid used?

Nassze Reply

how do lnternal energy measures

Esrael

Two bodies attract each other electrically. Do they both have to be charged? Answer the same question if the bodies repel one another.

JALLAH Reply

No. According to Isac Newtons law. this two bodies maybe you and the wall beside you. Attracting depends on the mass och each body and distance between them.

Dlovan

Are you really asking if two bodies have to be charged to be influenced by Coulombs Law?

Robert

like charges repel while unlike charges atttact

Raymond

What is specific heat capacity

Destiny Reply

Specific heat capacity is a measure of the amount of energy required to raise the temperature of a substance by one degree Celsius (or Kelvin). It is measured in Joules per kilogram per degree Celsius (J/kg°C).

AI-Robot

specific heat capacity is the amount of energy needed to raise the temperature of a substance by one degree Celsius or kelvin

ROKEEB

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

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