Hi, I’m Rob. Welcome to Math Antics.
We’ve already learned a little about how exponents and roots are used in Arithmetic,
and now it’s time to learn the basics of how they’re used in Algebra.
As you know, one of the main differences between Arithmetic and Algebra is that
Algebra involves unknowns values and variables.
In Arithmetic, you might have the exponent “4 squared”,
but in Algebra you’re more likely to see this exponent “X squared”.
And when it comes to roots, instead of seeing the “square root of 16”,
you might see, the “square root of X”.
Of course, one of the main goals in Algebra is to figure out what those unknown values are,
and we’re going to learn a bit about how to do that in a minute.
But first, we’re going to learn something about exponents by looking at an important pattern in Algebra.
It’s the pattern formed by the expression ‘x’ to the ’n’th power, where ’n’ is any integer.
In this expression, ‘x’ could be any number, but ’n’ can only be an integer.
And to keep things simple in this video, we’re only going to consider non-negative integers.
That is, we’ll limit ’n’ to be this set of numbers: 0, 1, 2, 3, and so on
If ’n’ is 0 then we have ‘x’ to the 0th power.
If ’n’ is 1 then we have ‘x’ to the 1st power.
If ’n’ is 2 then we have ‘x’ to the 2nd power (or “x squared”).
If ’n’ is 3 then we have ‘x’ to the 3rd power (or “x cubed”)
and we could keep on going with this pattern… ‘x’ to the 4th, ‘x’ to the 5th… to infinity.
Okay, but what do these exponents mean?
We’ll, ‘x’ squared is pretty easy to understand.
We know from our definition of exponents that “x squared” would be the same as ‘x’ times ‘x’.
We also know that ‘x’ cubed would be ‘x’ times ‘x’ times ‘x’.
And going up to higher values of ’n’ would just mean multiplying more ‘x’s together.
But what about ‘x’ to the 1st power?
Well, if ‘x’ to the 2nd power means multiplying 2 ‘x’s together,
then ‘x’ to the 1st power should mean multiplying one ‘x’ together, which sounds kinda funny when we say it like that.
But as you can see, that pattern makes sense.
‘x’ to the 1st power would just be ‘x’.
And that helps us see an important rule about exponents.
ANY number raised to the 1st power is just itself.
This rule (or property) is similar to the identity property of multiplication that says
ANY number multiplied by ‘1’ is just itself.
Okay, so ‘x’ to the 1st power makes sense, but what about ‘x’ to the 0th power?
Does that mean NO ‘x’s multiplied together?
That seems even stranger and the rule about the 0th power may surprise you…
It seems like ‘x’ to the 0th power should be zero, but it’s actually ‘1’!
…which will make a lot more sense if we modify our pattern a little.
Do you remember, that because of the identity property of multiplication,
there is always a factor of ‘1’ in ANY multiplication problem.
4 is the same as 1 × 4.
5 is the same as 1 × 5, and so on.
Well, that means we can also include a factor of ‘1’ in our pattern of exponents.
‘x’ to the 1st is 1 times ‘x’,
‘x’ to the 2nd is 1 times ‘x’ times ‘x’,
‘x’ to the 3rd is 1 times ‘x’ times ‘x’ times ‘x’, and so on.
And if we continue that pattern the other direction,
you see that there will be a ‘1’ left there, even when all the ‘x’s are gone.
So now you know another important rule about exponents:
ANY number raised to the 0th power is just ‘1’.
Knowing these rules about exponents is important in Algebra
and will help us when we talk about Polynomials in the next video.
But for the rest of this video,
we’re going to learn how to solve the some really basic algebraic equations that involve exponents and roots.
Let’s start off with this equation: the square root of x = 3.
How do we solve for ‘x’ in this equation?
In other words, how do we figure out the value of ‘x’ without just guessing the answer?
Well, we know that the key to solving an algebraic equation
is to get the unknown value all by itself on one side of the equal sign.
And you might be thinking that in this equation, the ‘x’ looks like it’s ALREADY by itself.
After all, there are no other numbers with it!
But getting an unknown by itself means we need to isolate it from any other numbers AND operators so that it’s completely by itself.
In this equation, that means we need to somehow get rid of the square root sign that the ‘x’ is under.
Ah Ha! …need to get rid of that pesky square root sign, do you?
Let’s see… I’ll just wave my magic wand and…
Hmmm… that usually works…
Ah… I know…
[Coughing]
Huh… this is gonna be harder than I thought!
One… Two…
Woah! Woah! Woah! That seems a bit extreme! And… it won’t even help!
I mean this is a MATH operation, and to get rid of a math operation, you need to use it’s INVERSE operation.
Uh… well… I was gonna try that next.
In the video called “Exponents and Square Roots”, we learned that exponents and roots are inverse operations.
If we want to undo an exponent, we need to use a root.
And if we want to undo a root, we need to use an exponent.
So in this equation, to undo the 2nd root (or square root) of ‘x’, we’re going to need to raise it to the 2nd power, or “square it”.
If we square the square root of ‘x’, those operations will cancel out and we’ll be left with just ‘x’.
But why does that work?
Well, you can see why it works if you remember what the square root of ‘x’ really means.
The the square root of ‘x’ is a number that we could multiply together twice to get ‘x’.
For example, the square root of 4 is 2 because if you multiply 2 × 2 you get 4.
So since the square root of 4 is the same as 2, we could also just say that the square root of 4 times the square root of 4 is 4.
And do you see how the square root of 4 times the square root of 4 is the same as the square root of 4 SQUARED?
And this is true for any number, which is why squaring the square root of ‘x’ just leaves us with ‘x’.
The exponent and the root operation cancel each other out.
Okay, so we can undo the square root by squaring that side of the equation,
but remember… to keep our equation in balance,
we need to do the same thing to both sides, so we need to square the 3 also.
3 squared is 3 × 3 which is 9.
There… by squaring BOTH sides of the equation, we changed it into x = 9. We solved for x.
That was pretty easy.
Let’s try solving another simple problem with a root.
This one is: the cube root of x = 5.
Just like before, we need to get ‘x’ all by itself by undoing the root,
but since it’s a cube root this time, we can’t undo that by squaring both sides.
Instead, we need to CUBE both sides.
You always need to undo a root with the corresponding exponent:
3rd root… 3rd power, 4th root… 4th power, and so on.
So to solve this equation, we need to raise each side of the equation to the 3rd power.
On the first side, the operations cancel, leaving ‘x’ all by itself,
and on the other side we have 5 to the 3rd power, which is 5 × 5 × 5 or 125. So x = 125.
Alright, so that’s how you solve very simple one-step equations with roots.
What about simple equations that have exponents instead of roots? …like this one: x squared = 36.
Again, we need to get the ‘x’ all by itself, which means we need to deal with the exponent on this side of the equation.
How do we undo an exponent?
Yep, we use a root!
Since the ‘x’ is being squared, if we take the square root of ‘x squared’, the operations will cancel out, leaving ‘x’ all by itself.
But why does that work?
Well, think for a minute about what the square root of ‘x squared’ would mean.
It means that you need to figure out what number you could multiply together twice in order to get ‘x squared’.
But that’s easy… ‘x’ times ‘x’ is ‘x squared’, so that means the square root of ‘x squared’ is just ‘x’.
So to solve this equation, we take the square root of BOTH sides of the equation (to keep things in balance)
On the first side, the operations cancel out leaving ‘x’ all by itself,
and on the other side, we have the square root of 36, which is 6.
So the answer to this problem is x = 6.
Well… that’s HALF of the answer anyway.
This problem is actually a little more complicated than it looks at first, thanks to negative numbers.
Do you remember in our video about multiplying and dividing integers?…
we learned that if you multiply two negative numbers together, the answer is actually POSITIVE.
That turns out to be really important when it comes to roots because it means there is often more than one answer.
For example, we know that the square root of 36 is 6, because multiplying 6 × 6 gives us 36.
But because of that rule about negative numbers, ‘negative 6’ times ‘negative 6’ is ALSO 36,
so it would be just as correct to say that the square root of 36 is ‘negative 6’.
So which is it? Is the square root of 36, 6 or -6?
The answer is both!
This is an example of a simple algebraic equation that has TWO solutions.
‘x’ could be 6… or ‘x’ could be -6.
‘x’ can’t be both 6 and -6 at the same time,
but you could substitute either value into the equation and it would make the equation true.
So in algebra, when we have a situation like this, where the answer could be positive OR negative,
we use a special “plus or minus sign” that looks like this. x = + or - 6.
And we use it when we are finding “even” roots of a number since we know the answer could be positive or negative.
But what about “odd” roots like the cube root of a number.
Like what if we have to solve the equation: x cubed = 27.
To solve this equation for ‘x’, we need to take the CUBE root of both sides.
On the first side of the equation, the cube root will cancel out the cube operation that’s being done to ‘x’, leaving ‘x’ all by itself.
And on the other side, we need to figure out the cube root of 27.
Using a calculator (or just by knowing about the factors of 27) we see that the cube root of 27 is 3, because 3 × 3 × 3 is 27.
So in this equation, we know that x = 3.
But what about negative numbers? Is x = -3 also a valid solution to this equation?
Nope! And here’s why.
If you multiply -3 times -3 times -3, the answer would be NEGATIVE 27, not 27.
So the cube root of 27 is 3 but NOT -3. In this case, there’s only one solution.
Alright, in this video, we learned two important rules about exponents.
We learned that ANY number raised to the 0th power equals ‘1’
and that ANY number raised to the 1st power is just itself.
We also learned how to solve very simple one-step equations involving exponents and roots.
Since they are inverse operations, to undo a root, you use its corresponding exponent,
and to undo an exponent, you use its corresponding root.
Of course, there’s a lot more to learn about exponents in algebra, but those are the basics.
And to make sure you really understand them, it’s important to practice by doing some exercise problems.
As always, thanks for watching Math Antics and I’ll see ya next time.
Learn more at www.mathantics.com