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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?

• 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.

Hi, I’m Rob. Welcome to Math Antics.

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# 代数学の基礎。代数学の指数 - Math Antics (Algebra Basics: Exponents In Algebra - Math Antics)

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Yassion Liu に公開 2021 年 01 月 14 日