**Learning Objectives**

After completing this tutorial, you should be able to:

- Solve radical equations.
- Solve equations that have rational exponents.

** Introduction**

In this tutorial, we will be looking at solving two different types
of equations, radical equations and equations that have rational exponents.
Both of these equations have the same ultimate goal, to get your variable
on one side and everything else on the other side using inverse operations.
Also, after removing the radical or rational exponent in the equations
in this tutorial, they become either a linear or quadratic equation.
Good news and bad news, as mentioned in other tutorials, a lot of
times in math you use previous knowledge to help work the new concepts.
That is good because you do not have to approach the problem as totally
new and learn all new steps. That can be overwhelming. It is
bad because you do have to remember things from the past. Sometimes
we condition ourselves to drain our brains after taking a test and sometimes
forget what we have learned. If you need a review on radicals in
general, feel free to go to **Tutorial
4: Radicals**. If you need a review on rational exponents in
general, feel free to go to **Tutorial
5: Rational Exponents**. If you need a review on solving linear
equations, feel free to go to **Tutorial
14: Linear Equations in One Variable**. If you need a review
on solving quadratic equations, feel free to go to **Tutorial
17: Quadratic Equations**. After going through this page,
you should be an old pro at working with roots. I think you are ready
to tackle these equations.

** Tutorial**

In other words, get one radical on one side and everything else on the
other using inverse operations.

In some problems there is only one radical. However, there are some problems that have more than one radical. In these problems make sure you isolate just one.

**Step 2**: **Get rid
of your radical sign. **

The inverse operation to a radical or a root is to raise it to an exponent.
Which exponent? Good question, it would be the exponent that matches
the index or root number on your radical.

In other words, if you had a square root, you would have to square it to get rid of it. If you had a cube root, you would have to cube it to get rid of it, and so forth.

You can raise both sides to the 2nd power, 10th power, hundredth power, etc. As long as you do the same thing to both sides of the equation, the two sides will remain equal to each other.

**Step 3**: **If you
still have a radical sign left, repeat steps 1 and 2.**

Sometimes you start out with two or more radicals in your equation.
If that is the case and you have at least one nonradical term, you will
probably have to repeat steps 1 and 2.

**Step 4**: **Solve the
remaining equation. **

The equations in this tutorial will lead to either a linear or a quadratic
equation.

If you need a review on solving linear equations, feel free to go to **Tutorial
14: Linear Equations in One Variable**.

If you need a review on solving quadratic equations, feel free to go
to **Tutorial 17: Quadratic Equations**.

**Step 5**: **Check
for extraneous solutions.**

When solving radical equations, extra solutions may come up when you
raise both sides to an even power. These extra solutions are called
extraneous solutions. **If a value is an extraneous
solution, it is not a solution to the original problem.**

In radical equations, you check for extraneous solutions by plugging
in the values you found back into the original problem. If the left side
does not equal the right side, then you have an extraneous solution.

**Example
1**: Solve the radical equation .

The radical in this equation is already isolated.

If you square a square root, it will disappear. This is what
we want to do here so that we can get *x* out
from under the square root and continue to solve for it.

No more radicals exist, so we do not have to repeat steps 1 and 2.

In this example, the equation that resulted from squaring both sides
turned out to be a **linear equation**.

If you need a review on solving linear equations, feel free to go to **Tutorial
14: Linear Equations in One Variable**.

***Inverse of mult. by 2 is div. by 2**

***True statement**

**There is one solution to this radical equation: x = 22.**

**Example
2**: Solve the radical equation .

***Square root is by itself on one side of eq.**

If you square a square root, it will disappear. This is what
we want to do here so that we can get *x* out
from under the square root and continue to solve for it.

***Left side is a binomial
squared**

No more radicals exist, so we do not have to repeat steps 1 and 2.

In this example, the equation that resulted from squaring both sides
turned out to be a **quadratic** **equation**.

If you need a review on solving quadratic equations, feel free to go
to **Tutorial 17: Quadratic Equations.**

***Use Zero-Product
Principle**

***Set 1st factor = 0 and solve**

***Set 2nd factor = 0 and solve**

***False statement**

***True statement**

Since we got a true statement, *x* = 2 is
a solution.

**There is only one solution to this radical equation: x = 2.**

**Example
3**: Solve the radical equation .

***One square root is by itself on one side of
eq.**

If you square a square root, it will disappear. This is what
we want to do here so that we can get *y* out
from under the square root and continue to solve for it.

***Square root is by itself on one side of eq.**

***Inverse of taking the sq. root is squaring
it**

***Left side is a binomial
squared**

In this example, the equation that resulted from squaring both sides
turned out to be a **quadratic** **equation**.

If you need a review on solving quadratic equations, feel free to go
to **Tutorial 17: Quadratic Equations.**

***Quad. eq. in standard form**

***Factor
the trinomial**

***Use Zero-Product
Principle**

***Set 1st factor = 0 and solve**

***Set 2nd factor = 0 and solve**

***True statement**

***True statement**

**There are two solutions to this radical equation: y = 3 and y = -1.**

**AND can be written in the form**

In other words get the base with the rational exponent on one side
and everything else on the other using inverse operations.

**Step 2**: **Get rid
of the rational exponent. **

The inverse operation to a rational exponent is to raise it to the
reciprocal of that exponent.

This will get rid of the rational exponent on the expression.

This will get rid of the rational exponent on the expression.

For example, if the rational exponent is 2/3, then the inverse operation is to raise both sides to the 3/2 power.

You can raise both sides to ANY power. As long as you do the same thing to both sides of the equation, the two sides will remain equal to each other.

**Step 3**: **Solve the
remaining equation. **

The equations in this tutorial will lead to either a linear or a quadratic
equation.

If you need a review on solving linear equations, feel free to go to **Tutorial
14: Linear Equations in One Variable**.

If you need a review on solving quadratic equations, feel free to go
to **Tutorial 17: Quadratic Equations**.

**Step 4**: **Check
for extraneous solutions.**

When solving equations with rational exponents, extra solutions may
come up when you raise both sides to an even power. These extra solutions
are called extraneous solutions. **If a value
is an extraneous solution, it is not a solution to the original problem.**

In equations with rational exponents you check for extraneous solutions
by plugging in the values you found back into the original problem. If
the left side does not equal the right side than you have an extraneous
solution.

**Example
4**: Solve the rational exponent equation .

The base with the rational exponent is already isolated.

If you raise an expression that has a rational exponent to the reciprocal
of that rational exponent, the exponent will disappear. This
is what we want to do here so that we can get *x* out from under the rational exponent and continue to solve for it.

***Use def.
of rat. exp**

In this example the equation that resulted from raising both sides
to the 2/3 power turned out to be a **linear equation**.

**Tutorial
14: Linear Equations in One Variable**.

***Inverse of mult. by 3 is div. by 3**

***True statement**

**There is one solution to this rational exponent equation: x = 5.**

**Example
5**: Solve the rational exponent equation .

***Inverse of mult. by 2 is div. by 2**

***rat. exp. expression is by itself on one side
of eq.**

If you raise an expression that has a rational exponent to the reciprocal
of that rational exponent, the exponent will disappear. This
is what we want to do here so that we can get *x* out from under the rational exponent and continue to solve for it.

In this example, the equation that resulted from raising both sides
to the 3/5 power turned out to be a **linear equation**.

Also note that it is already solved for *x*.
So, we do not have to do anything on this step for this example.

***Plugging in 5 to the 3/5 power for x**

***True statement**

**There is one solution to this rational exponent equation: .**

**Example
6**: Solve the rational exponent equation .

***rat. exp. expression is by itself on one side
of eq.**

If you raise an expression that has a rational exponent to the reciprocal
of that rational exponent, the exponent will disappear. This
is what we want to do here so that we can get *x* out from under the rational exponent and continue to solve for it.

In this example, the equation that resulted from squaring both sides
turned out to be a **quadratic** **equation**.

If you need a review on solving quadratic equations, feel free to go
to **Tutorial 17: Quadratic Equations.**

***Quad. eq. in standard form**

***Factor
the trinomial**

***Use Zero-Product
Principle**

***Set 1st factor = 0 and solve**

***Set 2nd factor = 0 and solve**

***False statement**

***False statement**

**There is no solution to this rational exponent equation.**

** Practice Problems**

These are practice problems to help bring you to the next level.
It will allow you to check and see if you have an understanding of these
types of problems. **Math works just like anything
else, if you want to get good at it, then you need to practice it.
Even the best athletes and musicians had help along the way and lots of
practice, practice, practice, to get good at their sport or instrument.**
In fact there is no such thing as too much practice.

To get the most out of these, **you should work the problem out on
your own and then check your answer by clicking on the link for the answer/discussion
for that problem**. At the link you will find the answer
as well as any steps that went into finding that answer.

Practice Problems 1a - 1b:Solve each radical equation.

Practice Problems 2a - 2b:Solve each rational exponent equation.

** Need Extra Help on these Topics?**

**http://www.sosmath.com/algebra/solve/solve0/solve0.html#radical**

Problems 1,2, 3, & 4 of this part of the webpage helps you with
solving equations with radicals. **ONLY do
problems 1, 2, 3, & 4**.

**Go to Get Help Outside the
Classroom found in Tutorial 1: How to Succeed in a Math Class for some
more suggestions.**

Videos at this site were created and produced by Kim Seward and Virginia Williams Trice.

Last revised on Dec. 16, 2009 by Kim Seward.

All contents copyright (C) 2002 - 2010, WTAMU and Kim Seward.
All rights reserved.