Solving Radical Equations
Objective
So, then what are roots?
A root is our way to undo an exponent.
An exponent tells you how many times you need to multiply a number to get another number.
For example:
Is really just a shorter way of writing:
Now, a root is the opposite of that.
A root tells us how many times a number needs to be multiplied to become the number on the inside.
This sounds really complicated, but let’s use our example.
We know 4 to the 4th power is 256.
So let’s do the opposite, or take the 4th root of 256:
What this is asking is:
Which we know is actually:
But, since we only need one of them as an answer, our answer would be:
The rule for even square roots
So, technically what we just got is true.
However, an unwritten rule for even roots is to also include its negative component.
So, for our last example:
Even though we just proved that it is equal to:
It’s also equal to:
But that makes sense right?
If you’re not sure, let’s try it out.
So, as we can see, -4 is also a fourth root of 256.
Explanation of roots importance
So, roots are important because they are the inverse operation to exponential expressions.
Remember, an inverse operation is the opposite operation to another operation.
An example of this is subtraction.
The opposite of subtraction is….
Addition.
So they are inverse operations.
Same with multiplication. The inverse to multiplication is…..
Division.
So what’s the inverse operation to exponential expressions?
Roots.
SO WHAT DOES THIS ALL MEAN?
9,-9 = x
So what did we just do there?
We took the root of each side of the equation (because what you do to one side, you need to do to another).
A FEW RULES TO CONSIDER
Multiplying roots
You can multiply roots together, but only multiply.
An example of this is:
We know this works because we can also just solve the root problems and then multiply.
So:
You can also add roots together, however only if they are in the root.
Example:
Which is not the same as:
You can also divide roots together if you need to.
Example:
We know this works because we can also just solve the root problems and then divide.
So:
= 9
= 9
= 10, -10
= 6 + 8
= 14
YOU CAN ALSO BREAK SQUARE ROOTS
SIMPLIFIED PROPERLY
It’s also proper to not leave radicals in the denominator of a fraction.
An example is:
(Since having the same number in the numerator and denominator is actually 1)
So we have:
NEGATIVE EXPONENTS
WHAT ABOUT TAKING AN EXPONENT, TO AN EXPONENT?
Okay, so how do we solve?
So, to be honest, this is one of those problems that are way easier to show than explain.
So without further ado:
Example:
Let’s say we have an equation like:
Now, we know how to solve these types of equations.
We have to cube the function to get rid of the cubic root:
But we know that a is actually to the first power.
So let’s put it that way:
Now, we know we have the same base, so now we can set the powers equal to each other.
So, that leaves us with:
And we know how to solve this:
Finally, what we are left with is:
So…….what does this mean?
Fractional Exponents
What we just proved was that when a number is taken to a fractional exponent, it’s the same as taking the root of the number.
So, for example:
And so on…
Wow, that was a lot of review
But now we can use what we’ve learned to simplify radical expressions.
To be honest, this is something that is easier to show than it is to explain, so let’s start off with an example:
Example :
Simplify the expression:
We know that if we have an exponent inside another exponent, that we multiply the exponents together.
So let’s start with that:
Then our new expression is:
To make our lives easier though, we can also simplify our fraction to:
We also know that if we have a negative exponent, we need to flip the fraction, and then change the exponent to positive.
So, that leaves us with:
And of course, we change the fractional exponent to the radical:
Which simplifies to:
Okay, so what else can we do with radicals?
Well, to be honest, nothing is really changing from the last lesson, except that we’ve been dealing with expressions, and now we’re going to be dealing with equations instead.
So what’s the difference?
Well, basically one has an equal sign (equation), and the other doesn’t.
So, using what we know, let’s start with some examples:
Using what we know for equations
So, essentially we actually aren’t changing anything.
All we are doing is using the same rules as we have been, but in this case, also adding the extra steps we need to solve equations.�Again, this sounds way more difficult and complicated then it is.
So, let’s start with a simple one:
Example 1:
Solve the following:
Just like any other equation, this time around we want to incorporate SADMEP to make solving much easier.
So, using SADMEP
Now that we’ve used the SA in SADMEP, what’s left is the DMEP.
SA
DM
Now what we are left with is a radical.
And we know how to undo those!
We need to take the equation to the proper power.
In this case, the second power.
So:
And what’s left is:
Now, we use SADMEP again
SA
And we have:
Now that we’ve used the SA in SADMEP, what’s left is the DMEP.
DM
And finally we are left with:
WELCOME TO ALGEBRA 2!
Yeah, that was a lot of work, but that’s how it goes.
So to recap:
So, let’s look at a few more to help understand how to solve these.
EXAMPLE 2:
Solve the following:
Just like any other equation, this time around we want to incorporate SADMEP to make solving much easier.
So, using SADMEP
Now that we’ve used the SA in SADMEP, what’s left is the DMEP.
SA
DM
Now what we are left with is a radical.
And we know how to undo those!
We need to take the equation to the proper power.
In this case, the second power.
So:
And what’s left is:
Now, we use SADMEP again
SA
And we have:
Now that we’ve used the SA in SADMEP, what’s left is the DMEP.
DM
And finally we are left with:
EXAMPLE 3:
Solve the following:
This one is a little different, but we know how to solve this.
Basically, we have two radicals on each side.
So the way we start this is by taking the proper power immediately
Now what we are left with is a typical equation.
And what’s left is:
So now we use SADMEP to get the final answer.
So:
EXAMPLE 4:
Solve the following:
This one is a little different, but we know how to solve this.
Basically, we have two radicals on each side.
So the way we start this is by taking the proper power immediately
Now what we are left with is a typical equation.
And what’s left is:
So now we use SADMEP to get the final answer.
So:
Now let’s try some harder ones
It may seem like these should be as tough as they get.
However, what we find happens sometimes, is that we get an answer that may not be correct.
So, to make sure we have the right answer, we need to plug our answers back into the original equation.
For example:
Example 5:
Solve the equation. Check for extraneous solutions:
So, for this equation, we need to solve it like we would normally first
Then check our answers.
So:
Now we need check to make sure these answers are legit.
To do that, we need to plug in our answers.
So:
So this one works.
Now, let’s try our second answer:
This one doesn’t works.
So we know our answer is only 6.
Example 6:
Solve the equation. Check for extraneous solutions:
So, for this equation, we need to solve it like we would normally first
Then check our answers.
So:
Normally, we would just plug our answers back in to check
But let’s look at what we have.
We have x = -3, which is a viable answer.
Now, we know that there is an answer for this.
But is it a real answer or imaginary?
Since it’s an imaginary answer, it’s not our true solution.
So our answer is: x = -3
So moral of the story
CHECK YOUR ANSWERS!
Make sure to plug your answers back into the original equation to make sure your answer is correct.