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  • Hi, I’m Rob. Welcome to Math Antics.

  • Weve already learned a little about how exponents and roots are used in Arithmetic,

  • and now it’s time to learn the basics of how theyre 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 youre more likely to see this exponent “X squared”.

  • And when it comes to roots, instead of seeing thesquare root of 16”,

  • you might see, thesquare root of X”.

  • Of course, one of the main goals in Algebra is to figure out what those unknown values are,

  • and were going to learn a bit about how to do that in a minute.

  • But first, were 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, were only going to consider non-negative integers.

  • That is, well 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 5thto infinity.

  • Okay, but what do these exponents mean?

  • Well, ‘x’ squared is pretty easy to understand.

  • We know from our definition of exponents that “x squaredwould 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,

  • were 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

  • Hmmmthat usually works

  • Ah… I know

  • [Coughing]

  • Huhthis is gonna be harder than I thought!

  • OneTwo

  • Woah! Woah! Woah! That seems a bit extreme! Andit 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.

  • Uhwell… I was gonna try that next.

  • In the video calledExponents 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’, were going to need to raise it to the 2nd power, orsquare it”.

  • If we square the square root of ‘x’, those operations will cancel out and well 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 rememberto 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.

  • Thereby 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 squaredwould 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 squaredis 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.

  • Wellthat’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’ timesnegative 6’ is ALSO 36,

  • so it would be just as correct to say that the square root of 36 isnegative 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 specialplus or minus signthat looks like this. x = + or - 6.

  • And we use it when we are findingevenroots of a number since we know the answer could be positive or negative.

  • But what aboutoddroots 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

Hi, I’m Rob. Welcome to Math Antics.

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Algebra Basics: Exponents In Algebra - Math Antics

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    Yassion Liu   posted on 2016/07/22
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