Simplifying Radical Expressions
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?
ANOTHER EXAMPLE
LAST EXAMPLE
And that’s how you deal with exponents inside exponents
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…
Simplifying with Fractional Exponents
So now that we know what the denominator of a fractional exponent means, we need to go over what the numerator means.
To do this, we need to do some 4th grade math.
REVIEW OVER MULTIPLYING FRACTIONS
To start, this is not an insult to your intelligence.
This is a genuine review because this is math you may not have done for a long while.
So to review, how do we multiply fractions?
We multiply the numerators together
Then we multiply the denominators together.
So, if we have something like:
We multiply the numerators together
Then we multiply the denominators together
SO WHY DO WE NEED TO KNOW THIS?
Because this will let us undo problems we may have, to help us solve them easier.
For example, let’s say we have something like:
We know that our fraction can be broken up into:
So, now let’s apply this to our problem:
Which is the same as saying:
Now, following PEMDAS, we:
125
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 1:
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:
Example 2:
Simplify the expression:
First we know that if we multiply exponents, we need to add them together.
So, let’s start with that:
Adding the fractions together, we can see that we have:
Now we see we have division.
Which means we need to subtract the exponents.
So, now we are left with:
And if we simplify, we can see we have:
Which is actually just:
Example 3:
Simplify the following:
To start off, we have no clue how to multiply these together.
But, what we can do is convert the radicals into fractional exponents.
Now, we can’t really multiply these because we have different bases.
However, maybe there’s a way we can get the same base?
We know that 49 is 7 squared, so what if we replaced 49 with 7 squared?
Well, we know how to solve an exponent inside of an exponent:
And now that we have the same base, we can add.
However, since we don’t have the same denominator, we need to find it first.
So:
Which simplifies to:
So now we have:
And finally, adding them together, we get:
Example 4:
Simplify the following:
So, like the other examples, let’s first convert the radicals to fractional exponents.
And we know we can simplify to:
So, now all we need to do is subtract:
And our answer is:
EXAMPLE 5:
Simplify the following:
As per the usual, let’s first convert the radicals to fractional exponents.
And now we add the fractions:
Now we subtract the fractions:
And we have:
Finally, applying the rule of negative exponents, we have:
It may be ugly, but there is our answer.