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So you have 2f minus 1 times this 3f, times that 3f, and then times that plus 11. Let me do that in the same shade of purple right over there. And you can distribute, if you like. 2f minus 1 times 3f will give you this term. 2f minus 1 times 11 will give you that term. And we can't forget that we still have that negati... | Example 4 Factoring quadratics by taking a negative common factor and grouping Khan Academy.mp3 |
2f minus 1 times 3f will give you this term. 2f minus 1 times 11 will give you that term. And we can't forget that we still have that negative 2 hanging out outside. I don't want to change the colors on it. We have the negative 2 hanging out, that same negative 2 over there. And we're done factoring it. Negative 12f sq... | Example 4 Factoring quadratics by taking a negative common factor and grouping Khan Academy.mp3 |
Four x minus one is equal to three y plus five. Now when we look at an ordered pair and we want to figure out whether it's a solution, we just have to remind ourselves that in these ordered pairs, the convention, the standard is, is that the first coordinate is the x coordinate and the second coordinate is the y coordi... | Checking ordered pair solutions to equations example 2 Algebra I Khan Academy.mp3 |
So let's try that out. So we have four times x. Well, we're saying x needs to be equal to three minus one is going to be equal to three times y. Well, if this ordered pair is a solution, then y is going to be equal to two. So three times y, y is two, plus five. Notice all I did is wherever I saw the x, I substituted it... | Checking ordered pair solutions to equations example 2 Algebra I Khan Academy.mp3 |
Well, if this ordered pair is a solution, then y is going to be equal to two. So three times y, y is two, plus five. Notice all I did is wherever I saw the x, I substituted it with three, wherever I saw the y, I substituted it with two. Now let's see if this is true. Four times three is 12 minus one. Is this really the... | Checking ordered pair solutions to equations example 2 Algebra I Khan Academy.mp3 |
Now let's see if this is true. Four times three is 12 minus one. Is this really the same thing as three times two, which is six plus five? See, 12 minus one is 11. Six plus five is also 11. This is true. 11 equals 11. | Checking ordered pair solutions to equations example 2 Algebra I Khan Academy.mp3 |
See, 12 minus one is 11. Six plus five is also 11. This is true. 11 equals 11. This pair, three comma two, does satisfy this equation. Now let's see whether this one does. Two comma three. | Checking ordered pair solutions to equations example 2 Algebra I Khan Academy.mp3 |
11 equals 11. This pair, three comma two, does satisfy this equation. Now let's see whether this one does. Two comma three. So this is saying when x is equal to two, y would be equal to three for this equation. Let's see if that's true. So four times x, we're now going to see if when x is two, y can be three. | Checking ordered pair solutions to equations example 2 Algebra I Khan Academy.mp3 |
Two comma three. So this is saying when x is equal to two, y would be equal to three for this equation. Let's see if that's true. So four times x, we're now going to see if when x is two, y can be three. So four times x, four times two, minus one is equal to three times y. Now y we're testing to see if it can be three.... | Checking ordered pair solutions to equations example 2 Algebra I Khan Academy.mp3 |
So four times x, we're now going to see if when x is two, y can be three. So four times x, four times two, minus one is equal to three times y. Now y we're testing to see if it can be three. Three times three plus five. Let's see if this is true. Four times two is eight minus one. Is this equal to three times three? | Checking ordered pair solutions to equations example 2 Algebra I Khan Academy.mp3 |
Three times three plus five. Let's see if this is true. Four times two is eight minus one. Is this equal to three times three? So that's nine plus five. So is seven equal to 14? No, clearly seven is not equal to 14. | Checking ordered pair solutions to equations example 2 Algebra I Khan Academy.mp3 |
Is this equal to three times three? So that's nine plus five. So is seven equal to 14? No, clearly seven is not equal to 14. So these things are not equal to each other. So this is not a solution. When x equals two, y cannot be equal to three and satisfy this equation. | Checking ordered pair solutions to equations example 2 Algebra I Khan Academy.mp3 |
And like always, pause the video and try to work it out before I do. Well, when you look at this, we have these two rational expressions and we have the same denominator, two x squared minus seven. So you could say we have six two x squared minus sevenths and then we have negative three x minus eight two x squared minu... | Adding & subtracting rational expressions like denominators High School Math Khan Academy.mp3 |
So if you have the same denominator, this is going to be equal to, this is going to be equal to, our denominator's going to be two x squared minus seven, two x squared minus seven, and then we just add the numerators. So it's going to be six plus negative three x, negative three x minus eight. And so if we want to simp... | Adding & subtracting rational expressions like denominators High School Math Khan Academy.mp3 |
Six plus negative eight is going to be negative two, so it's going to be negative two, and then adding a negative three x, that's the same thing as subtracting three x, so negative two minus three x, all of that over, all of that, do that same blue color, all of that over two x squared minus seven. And we're done, we'v... | Adding & subtracting rational expressions like denominators High School Math Khan Academy.mp3 |
So here, we want to subtract one rational expression from another. So see if you can figure that out. Well, once again, both of these rational expressions have the exact same denominator. The denominator for both of them is 14 x squared minus nine, 14 x squared minus nine, so the denominator of the difference, I guess ... | Adding & subtracting rational expressions like denominators High School Math Khan Academy.mp3 |
The denominator for both of them is 14 x squared minus nine, 14 x squared minus nine, so the denominator of the difference, I guess we can call it that, is going to be 14 x squared minus nine. So 14 x squared minus nine. Did I say four x squared before? 14 x squared minus nine, that's the denominator of both of them, s... | Adding & subtracting rational expressions like denominators High School Math Khan Academy.mp3 |
14 x squared minus nine, that's the denominator of both of them, so that's going to be the denominator of our answer right over here. And so, we can just subtract the numerators. So we're gonna have nine x squared plus three minus, minus all of this business, minus negative three x squared plus five, and so we can dist... | Adding & subtracting rational expressions like denominators High School Math Khan Academy.mp3 |
This is going to be equal to, this is going to be equal to nine x squared plus three, and then if you distribute the negative sign, the negative of negative three x squared is going to be plus three x squared, and then the negative of positive five is going to be negative five, so we're gonna subtract five from that, a... | Adding & subtracting rational expressions like denominators High School Math Khan Academy.mp3 |
In the following graph is y a function of x. So in order for y to be a function of x, for any x that you input into the function, any x for which the function is defined, so let's say we have y is equal to f of x, so we have our little function machine, it should spit out exactly one value of y. If it spits out multipl... | Does a vertical line represent a function Functions and their graphs Algebra II Khan Academy.mp3 |
It could be equal to any of those possible values for y. So let's see if this, for this graph, whether for a given x it spits out exactly one y. Well, the function seems to be only defined, so the domain of this function is x is equal to negative two. That's the only place where we have a definition for it. And if we t... | Does a vertical line represent a function Functions and their graphs Algebra II Khan Academy.mp3 |
That's the only place where we have a definition for it. And if we try to input negative two into this little black box, what do we get? Do we get exactly one thing? No, we put in negative two here. We could get anything. Negative two, the point negative two nine is on this relation. Negative two eight is on this relat... | Does a vertical line represent a function Functions and their graphs Algebra II Khan Academy.mp3 |
No, we put in negative two here. We could get anything. Negative two, the point negative two nine is on this relation. Negative two eight is on this relation. Negative two seven, negative two 7.5, negative two 3.14159, they're all on these. So if you put a negative two into this relation, you actually get, essentially,... | Does a vertical line represent a function Functions and their graphs Algebra II Khan Academy.mp3 |
We're asked to divide, and we're dividing 6 plus 3i by 7 minus 5i. And in particular, when I divide this, I want to get another complex number. So I want to get something, you know, some real number plus some imaginary number. So some multiple of i. So let's think about how we can do this. Well, division is the same th... | Dividing complex numbers Imaginary and complex numbers Precalculus Khan Academy.mp3 |
So some multiple of i. So let's think about how we can do this. Well, division is the same thing. We could rewrite this as 6 plus 3i over 7 minus 5i. These are clearly equivalent. Dividing by something is the same thing as a rational expression, where that something is in the denominator right over here. And so how do ... | Dividing complex numbers Imaginary and complex numbers Precalculus Khan Academy.mp3 |
We could rewrite this as 6 plus 3i over 7 minus 5i. These are clearly equivalent. Dividing by something is the same thing as a rational expression, where that something is in the denominator right over here. And so how do we simplify this? Well, we have a tool in our toolkit that can make sure that we don't have an ima... | Dividing complex numbers Imaginary and complex numbers Precalculus Khan Academy.mp3 |
And so how do we simplify this? Well, we have a tool in our toolkit that can make sure that we don't have an imaginary or a complex number in the denominator. And that's the complex conjugate. If we multiply both the numerator and the denominator of this expression by the complex conjugate of the denominator, then we w... | Dividing complex numbers Imaginary and complex numbers Precalculus Khan Academy.mp3 |
If we multiply both the numerator and the denominator of this expression by the complex conjugate of the denominator, then we will get rid of, or we will have a real number in the denominator. So let's do that. Let's multiply the numerator and the denominator by the conjugate of this. So 7 plus 5i. Plus 7 plus 5i is th... | Dividing complex numbers Imaginary and complex numbers Precalculus Khan Academy.mp3 |
So 7 plus 5i. Plus 7 plus 5i is the complex conjugate of 7 minus 5i. So we're going to multiply it by 7 plus 5i over 7 plus 5i. And anything divided by itself is going to be 1, assuming that you're not dealing with 0. 0 over 0 is undefined. But 7 plus 5i over 7 plus 5i is 1. So we're not changing the value of this. | Dividing complex numbers Imaginary and complex numbers Precalculus Khan Academy.mp3 |
And anything divided by itself is going to be 1, assuming that you're not dealing with 0. 0 over 0 is undefined. But 7 plus 5i over 7 plus 5i is 1. So we're not changing the value of this. But what this does is it allows us to get rid of the imaginary part in the denominator. So let's multiply this out. Our numerator, ... | Dividing complex numbers Imaginary and complex numbers Precalculus Khan Academy.mp3 |
So we're not changing the value of this. But what this does is it allows us to get rid of the imaginary part in the denominator. So let's multiply this out. Our numerator, we just have to multiply every part of this complex number times every part of this complex number. You can think of it as foil if you like. We're r... | Dividing complex numbers Imaginary and complex numbers Precalculus Khan Academy.mp3 |
Our numerator, we just have to multiply every part of this complex number times every part of this complex number. You can think of it as foil if you like. We're really just doing the distributive property twice. We have 6 times 6 is, or sorry, 6 times 7, which is 42. And then we have 6 times 5i, which is 30i plus 30i.... | Dividing complex numbers Imaginary and complex numbers Precalculus Khan Academy.mp3 |
We have 6 times 6 is, or sorry, 6 times 7, which is 42. And then we have 6 times 5i, which is 30i plus 30i. And then we have 3i times 7. So that's plus 21i. And then finally, we have 3i times 5i. 3 times 5 is 15. But we have i times i, or i squared, which is negative 1. | Dividing complex numbers Imaginary and complex numbers Precalculus Khan Academy.mp3 |
So that's plus 21i. And then finally, we have 3i times 5i. 3 times 5 is 15. But we have i times i, or i squared, which is negative 1. So it would be 15 times negative 1, or minus 15. So that's our numerator. And then our denominator is going to be, well, we have a plus b times a minus b. | Dividing complex numbers Imaginary and complex numbers Precalculus Khan Academy.mp3 |
But we have i times i, or i squared, which is negative 1. So it would be 15 times negative 1, or minus 15. So that's our numerator. And then our denominator is going to be, well, we have a plus b times a minus b. You could think of it that way, or you could just do what we just did up here. Actually, let's just do what... | Dividing complex numbers Imaginary and complex numbers Precalculus Khan Academy.mp3 |
And then our denominator is going to be, well, we have a plus b times a minus b. You could think of it that way, or you could just do what we just did up here. Actually, let's just do what we did up here so you don't have to remember that difference of squares pattern and all of that. 7 times 7 is 49. Let's think of it... | Dividing complex numbers Imaginary and complex numbers Precalculus Khan Academy.mp3 |
7 times 7 is 49. Let's think of it in a foil way if that is helpful for you. So first, we did the 7 times a 7, then we could do the outer terms. 7 times 5i is plus 35i. Then we could do the inner terms. Negative 5i times 7 is minus 35i. These two are going to cancel out. | Dividing complex numbers Imaginary and complex numbers Precalculus Khan Academy.mp3 |
7 times 5i is plus 35i. Then we could do the inner terms. Negative 5i times 7 is minus 35i. These two are going to cancel out. And then negative 5i times 5i is negative 25i squared. Negative 25i squared is the same thing as negative 25 times negative 1. So that is plus 25. | Dividing complex numbers Imaginary and complex numbers Precalculus Khan Academy.mp3 |
These two are going to cancel out. And then negative 5i times 5i is negative 25i squared. Negative 25i squared is the same thing as negative 25 times negative 1. So that is plus 25. Now let's simplify them. These guys down here cancel out. Our denominator simplifies to 49 plus 25 is 74. | Dividing complex numbers Imaginary and complex numbers Precalculus Khan Academy.mp3 |
So that is plus 25. Now let's simplify them. These guys down here cancel out. Our denominator simplifies to 49 plus 25 is 74. And our numerator, we can add the real parts. So we have a 42 and a negative 15. So let's see, 42 minus 5 would be 37, minus another 10 would be 27. | Dividing complex numbers Imaginary and complex numbers Precalculus Khan Academy.mp3 |
Our denominator simplifies to 49 plus 25 is 74. And our numerator, we can add the real parts. So we have a 42 and a negative 15. So let's see, 42 minus 5 would be 37, minus another 10 would be 27. So that is 27. And then we're going to add our 30i plus the 21i. So 30 of something plus 21 of that same something is going... | Dividing complex numbers Imaginary and complex numbers Precalculus Khan Academy.mp3 |
So let's see, 42 minus 5 would be 37, minus another 10 would be 27. So that is 27. And then we're going to add our 30i plus the 21i. So 30 of something plus 21 of that same something is going to be 51 of that something. In this case, that something is the imaginary unit. It is i. Let me do that in magenta. | Dividing complex numbers Imaginary and complex numbers Precalculus Khan Academy.mp3 |
So 30 of something plus 21 of that same something is going to be 51 of that something. In this case, that something is the imaginary unit. It is i. Let me do that in magenta. I guess that's orange. So this is plus 51i. And I want to write it in the form of a plus bi, the traditional complex number form. | Dividing complex numbers Imaginary and complex numbers Precalculus Khan Academy.mp3 |
Let me do that in magenta. I guess that's orange. So this is plus 51i. And I want to write it in the form of a plus bi, the traditional complex number form. So this right over here is the same thing as 27 over 74 plus 51 over 74 times i. And we are done. I want to write that i in that same orange color. | Dividing complex numbers Imaginary and complex numbers Precalculus Khan Academy.mp3 |
And I want to write it in the form of a plus bi, the traditional complex number form. So this right over here is the same thing as 27 over 74 plus 51 over 74 times i. And we are done. I want to write that i in that same orange color. And we are done. We have a real part and we have an imaginary part. And if this last s... | Dividing complex numbers Imaginary and complex numbers Precalculus Khan Academy.mp3 |
I want to write that i in that same orange color. And we are done. We have a real part and we have an imaginary part. And if this last step just confuses you a little bit, remember, this is the same thing if it's helpful for you. We're essentially multiplying both of these terms times 1 over 74. We're dividing both of ... | Dividing complex numbers Imaginary and complex numbers Precalculus Khan Academy.mp3 |
Let's see if we can figure out what x plus three times x minus three is, and I encourage you to pause the video and see if you can work this out. Well, one way to tackle it is the way that we've always tackled when we multiply binomials is just apply the distributive property twice. So first, we could take this entire ... | Special products of the form (x+a)(x-a) Algebra I High School Math Khan Academy.mp3 |
So first, we can multiply it times this x, so that's going to be x times x plus three, and then we are going to multiply it times, we could say this negative three. So we could write minus three times, now that's going to be multiplied by x plus three again, x plus three, and then we apply the distributive property one... | Special products of the form (x+a)(x-a) Algebra I High School Math Khan Academy.mp3 |
And what does this simplify to? Well, we're gonna get x squared, and we have three x minus three x, so these two characters cancel out, and we are just left with x squared minus nine. And you might see a little pattern here. Notice, I added three and then I subtracted three, and I got this, I got the x squared, and the... | Special products of the form (x+a)(x-a) Algebra I High School Math Khan Academy.mp3 |
Notice, I added three and then I subtracted three, and I got this, I got the x squared, and then if you take three and multiply it by negative three, you are going to get a negative nine. And notice, the middle terms canceled out, and one thing you might ask is, well, will that always be the case if we add a number and... | Special products of the form (x+a)(x-a) Algebra I High School Math Khan Academy.mp3 |
Let's talk in general terms. So if we, instead of doing x plus three times x minus three, we could write the same thing as, instead of three, let's just say you have x plus, x plus a times x minus a, times x minus a. And I encourage you to pause this video and work it all out. Just assume a is some number, like three o... | Special products of the form (x+a)(x-a) Algebra I High School Math Khan Academy.mp3 |
Just assume a is some number, like three or some other number, and apply the distributive property twice and see what you get. Well, let's work through it. So first we can distribute this yellow x plus a onto the x and the negative a. So x plus a times x, or we could say x times x plus a. So that's going to be, that's ... | Special products of the form (x+a)(x-a) Algebra I High School Math Khan Academy.mp3 |
So x plus a times x, or we could say x times x plus a. So that's going to be, that's going to be x times x plus a, and then we're going to have minus a, or this negative a times x plus a. So minus, and then we're going to have this minus a times x plus a, times x plus a, times x plus a. Notice, all I did is I distribut... | Special products of the form (x+a)(x-a) Algebra I High School Math Khan Academy.mp3 |
Notice, all I did is I distributed this yellow, I just distributed this big chunk of this expression, I just distributed it onto the x and onto this negative a. I'm multiplying it times the x and I'm multiplying it by the negative a. And now we can apply the distributive property again. X times x is x squared. X times ... | Special products of the form (x+a)(x-a) Algebra I High School Math Khan Academy.mp3 |
X times a is ax. And then we get negative a times x is negative ax. And then negative a times a is negative a squared. And notice, regardless of my choice of a, I'm going to have ax and then minus ax. So this is always going to cancel out. It didn't just work for the case when a was three. For any a, if I have a times ... | Special products of the form (x+a)(x-a) Algebra I High School Math Khan Academy.mp3 |
And notice, regardless of my choice of a, I'm going to have ax and then minus ax. So this is always going to cancel out. It didn't just work for the case when a was three. For any a, if I have a times x and then I subtract a times x, that's just going to cancel out. So this is just going to cancel out. And what are we ... | Special products of the form (x+a)(x-a) Algebra I High School Math Khan Academy.mp3 |
For any a, if I have a times x and then I subtract a times x, that's just going to cancel out. So this is just going to cancel out. And what are we going to be left with? We are going to be left with x squared minus a squared. X squared minus a squared. And you could view this as a special case. When you have something... | Special products of the form (x+a)(x-a) Algebra I High School Math Khan Academy.mp3 |
We are going to be left with x squared minus a squared. X squared minus a squared. And you could view this as a special case. When you have something x plus something times x minus that same something, it's going to be x squared minus that something squared. And this is a good one to know in general. This is a good one... | Special products of the form (x+a)(x-a) Algebra I High School Math Khan Academy.mp3 |
When you have something x plus something times x minus that same something, it's going to be x squared minus that something squared. And this is a good one to know in general. This is a good one to know in general. And we could use it to quickly figure out the products of other binomials that fit this pattern here. So ... | Special products of the form (x+a)(x-a) Algebra I High School Math Khan Academy.mp3 |
And we could use it to quickly figure out the products of other binomials that fit this pattern here. So if I were to say, quick, what is x plus 10 times x minus 10? Well, you could say, all right, this fits the pattern. It's x plus a times x minus a. So it's going to be x squared minus a squared. If a is 10, a squared... | Special products of the form (x+a)(x-a) Algebra I High School Math Khan Academy.mp3 |
If you don't believe that one of these properties are true and you want them proved, I've made three or four videos that actually prove these properties. But what I'm going to do is I'm going to show you the properties and then show you how they can be used. It's going to be a little more hands-on. So let's just do a l... | Introduction to logarithm properties Logarithms Algebra II Khan Academy.mp3 |
So let's just do a little bit of a review of just what a logarithm is. So if I say that a, oh, that's not the right, let's see, I want to change, there you go. Let's say I say that a, let me start over, a to the b is equal to c. a to the b to the power is equal to c. So another way to write this exact same relationship... | Introduction to logarithm properties Logarithms Algebra II Khan Academy.mp3 |
So we could say that the logarithm base a of c is equal to b. So these are essentially saying the same thing, they just have different kind of results. In one, you know a and b and you're kind of getting c. That's what exponentiation does for you. And the second one, you know a and you know that when you raise it to so... | Introduction to logarithm properties Logarithms Algebra II Khan Academy.mp3 |
And the second one, you know a and you know that when you raise it to some power, you get c. And then you figure out what b is. So the exact same relationship, just dated in a different way. Now I will introduce you to some interesting logarithm properties. And they actually just fall out of this relationship and the r... | Introduction to logarithm properties Logarithms Algebra II Khan Academy.mp3 |
And they actually just fall out of this relationship and the regular exponent rules. So the first is that the logarithm, let me do a more cheerful color. The logarithm, let's say, of any base, so let's just call the base, let's say b for base, logarithm base b of a plus logarithm base b of c. And this only works if we ... | Introduction to logarithm properties Logarithms Algebra II Khan Academy.mp3 |
That equals the logarithm of base b of a times c. Now what does this mean and how can we use it? Or let's just even try it out with some, I don't know, examples. So this is saying that, I'll switch to another color. Let's make mauve my mauve. I don't know, I never know how to say that properly. Let's make that my examp... | Introduction to logarithm properties Logarithms Algebra II Khan Academy.mp3 |
Let's make mauve my mauve. I don't know, I never know how to say that properly. Let's make that my example color. So let's say logarithm of base 2 of, I don't know, of 8 plus logarithm base 2 of, I don't know, let's say 32. So in theory, this should equal, if we believe this property, this should equal logarithm base 2... | Introduction to logarithm properties Logarithms Algebra II Khan Academy.mp3 |
So let's say logarithm of base 2 of, I don't know, of 8 plus logarithm base 2 of, I don't know, let's say 32. So in theory, this should equal, if we believe this property, this should equal logarithm base 2 of what? Well we say 8 times 32. So 8 times 32 is 240 plus 16, 256. Let's see if that's true just trying out this... | Introduction to logarithm properties Logarithms Algebra II Khan Academy.mp3 |
So 8 times 32 is 240 plus 16, 256. Let's see if that's true just trying out this number. And this really isn't a proof, but it'll give you a little bit of an intuition, I think, for what's going on around here. So we just used our property, this little property that I presented to you, and let's see if it works out. So... | Introduction to logarithm properties Logarithms Algebra II Khan Academy.mp3 |
So we just used our property, this little property that I presented to you, and let's see if it works out. So log base 2 of 8. 2 to what power is equal to 8? Well 2 to the third power is equal to 8, right? 2 to the third power is equal to 8. So this term right here, that equals 3, right? Log base 2 of 8 is equal to 3. | Introduction to logarithm properties Logarithms Algebra II Khan Academy.mp3 |
Well 2 to the third power is equal to 8, right? 2 to the third power is equal to 8. So this term right here, that equals 3, right? Log base 2 of 8 is equal to 3. 2 to what power is equal to 32? Let's see, 2 to the fourth power is 16, 2 to the fifth power is 32. So this right here is 5, right? | Introduction to logarithm properties Logarithms Algebra II Khan Academy.mp3 |
Log base 2 of 8 is equal to 3. 2 to what power is equal to 32? Let's see, 2 to the fourth power is 16, 2 to the fifth power is 32. So this right here is 5, right? And 2 to the what power is equal to 256? Well, let's see. Well if you're a computer science major, you'll know that immediately. | Introduction to logarithm properties Logarithms Algebra II Khan Academy.mp3 |
So this right here is 5, right? And 2 to the what power is equal to 256? Well, let's see. Well if you're a computer science major, you'll know that immediately. That a byte can have 256 values in it. So it's 2 to the eighth power. But if you don't know that, you could multiply it out yourself. | Introduction to logarithm properties Logarithms Algebra II Khan Academy.mp3 |
Well if you're a computer science major, you'll know that immediately. That a byte can have 256 values in it. So it's 2 to the eighth power. But if you don't know that, you could multiply it out yourself. But this is 8. And I'm not doing it just because I knew that 3 plus 5 is equal to 8. I'm doing this independently. | Introduction to logarithm properties Logarithms Algebra II Khan Academy.mp3 |
But if you don't know that, you could multiply it out yourself. But this is 8. And I'm not doing it just because I knew that 3 plus 5 is equal to 8. I'm doing this independently. So this is equal to 8. But it does turn out that 3 plus 5 is equal to 8. This may seem like magic to you, or it may seem obvious. | Introduction to logarithm properties Logarithms Algebra II Khan Academy.mp3 |
I'm doing this independently. So this is equal to 8. But it does turn out that 3 plus 5 is equal to 8. This may seem like magic to you, or it may seem obvious. And for those of you who it might seem a little obvious, you're probably thinking, well, 2 to the third times 2 to the fifth is equal to 2 to the 3 plus 5, righ... | Introduction to logarithm properties Logarithms Algebra II Khan Academy.mp3 |
This may seem like magic to you, or it may seem obvious. And for those of you who it might seem a little obvious, you're probably thinking, well, 2 to the third times 2 to the fifth is equal to 2 to the 3 plus 5, right? This is just an exponent rule. What do they call this? The additive exponent? I don't know. I don't ... | Introduction to logarithm properties Logarithms Algebra II Khan Academy.mp3 |
What do they call this? The additive exponent? I don't know. I don't know the names of things. And that equals 2 to the eighth. And that's exactly what we did here, right? On this side, we had 2 to the third times 2 to the fifth, essentially. | Introduction to logarithm properties Logarithms Algebra II Khan Academy.mp3 |
I don't know the names of things. And that equals 2 to the eighth. And that's exactly what we did here, right? On this side, we had 2 to the third times 2 to the fifth, essentially. And on this side, you have them added to each other. And what makes logarithms interesting is, and why it's a little confusing at first, a... | Introduction to logarithm properties Logarithms Algebra II Khan Academy.mp3 |
On this side, we had 2 to the third times 2 to the fifth, essentially. And on this side, you have them added to each other. And what makes logarithms interesting is, and why it's a little confusing at first, and you can watch the proofs if you really want a kind of a rigorous, not even my proofs aren't rigorous, but if... | Introduction to logarithm properties Logarithms Algebra II Khan Academy.mp3 |
But this should hopefully give you an intuition for why this property holds, right? Because when you multiply two numbers of the same base, two exponential expressions of the same base, you can add their exponents. Similarly, when you have the log of two numbers multiplied by each other, that's equivalent to the log of... | Introduction to logarithm properties Logarithms Algebra II Khan Academy.mp3 |
This is the same property. If you don't believe me, watch the proof videos. So let me show you another log property that's pretty much the same one. I almost view them the same. So this is log base b of a minus log base b of c is equal to log base b of a divided by c. That says a divided by c. And we can, once again, t... | Introduction to logarithm properties Logarithms Algebra II Khan Academy.mp3 |
I almost view them the same. So this is log base b of a minus log base b of c is equal to log base b of a divided by c. That says a divided by c. And we can, once again, try it out with some numbers. I use 2 a lot, just because 2 is an easy number to figure out the powers. But let's use a different number. Let's say lo... | Introduction to logarithm properties Logarithms Algebra II Khan Academy.mp3 |
But let's use a different number. Let's say log base 3 of 1 ninth minus log base 3 of 81. So this property tells us that this is the same thing as, well, I'm ending up with a big number. Log base 3 of 1 ninth divided by 81. So that's the same thing as 1 ninth times 1 over 81. I used two large numbers for my example. Bu... | Introduction to logarithm properties Logarithms Algebra II Khan Academy.mp3 |
Log base 3 of 1 ninth divided by 81. So that's the same thing as 1 ninth times 1 over 81. I used two large numbers for my example. But we'll move forward. So let's see. 9 times 8 is 720. Right? | Introduction to logarithm properties Logarithms Algebra II Khan Academy.mp3 |
But we'll move forward. So let's see. 9 times 8 is 720. Right? 9 times 8 is 720. So this is 1 over 729. So this is log base 3 over 1 over 729. | Introduction to logarithm properties Logarithms Algebra II Khan Academy.mp3 |
Right? 9 times 8 is 720. So this is 1 over 729. So this is log base 3 over 1 over 729. So 3 to what power is equal to 1 ninth? Well, 3 squared is equal to 9, right? 3 squared is equal to 9. | Introduction to logarithm properties Logarithms Algebra II Khan Academy.mp3 |
So this is log base 3 over 1 over 729. So 3 to what power is equal to 1 ninth? Well, 3 squared is equal to 9, right? 3 squared is equal to 9. So we know that if 3 squared is equal to 9, then we know that 3 to the negative 2 is equal to 1 ninth, right? The negative just inverts it. So this is equal to negative 2. | Introduction to logarithm properties Logarithms Algebra II Khan Academy.mp3 |
3 squared is equal to 9. So we know that if 3 squared is equal to 9, then we know that 3 to the negative 2 is equal to 1 ninth, right? The negative just inverts it. So this is equal to negative 2. Right? And then minus 3 to what power is equal to 81? Let's see. | Introduction to logarithm properties Logarithms Algebra II Khan Academy.mp3 |
So this is equal to negative 2. Right? And then minus 3 to what power is equal to 81? Let's see. 3 to the third power is 27. So 3 to the fourth power. So we have minus 2 minus 4 is equal to, well, we could do it a couple of ways. | Introduction to logarithm properties Logarithms Algebra II Khan Academy.mp3 |
Let's see. 3 to the third power is 27. So 3 to the fourth power. So we have minus 2 minus 4 is equal to, well, we could do it a couple of ways. Minus 2 minus 4 is equal to minus 6. And now we just have to confirm that 3 to the minus 6 power is equal to 1 over 729. So my question is, 3 to the minus 6 power, is that equa... | Introduction to logarithm properties Logarithms Algebra II Khan Academy.mp3 |
So we have minus 2 minus 4 is equal to, well, we could do it a couple of ways. Minus 2 minus 4 is equal to minus 6. And now we just have to confirm that 3 to the minus 6 power is equal to 1 over 729. So my question is, 3 to the minus 6 power, is that equal to 1 over 729? Well, that's the same thing as saying 3 to the 6... | Introduction to logarithm properties Logarithms Algebra II Khan Academy.mp3 |
So my question is, 3 to the minus 6 power, is that equal to 1 over 729? Well, that's the same thing as saying 3 to the 6th power is equal to 729, because that's all the negative exponent does, is inverts it. Let's see. We could multiply that out, but that should be the case. Because, well, we could look here, but let's... | Introduction to logarithm properties Logarithms Algebra II Khan Academy.mp3 |
We could multiply that out, but that should be the case. Because, well, we could look here, but let's see. 3 to the third power. This would be 3 to the third power times 3 to the third power is equal to 27 times 27. That looks pretty close. You can confirm it with a calculator if you don't believe me. Anyway, that's al... | Introduction to logarithm properties Logarithms Algebra II Khan Academy.mp3 |
This would be 3 to the third power times 3 to the third power is equal to 27 times 27. That looks pretty close. You can confirm it with a calculator if you don't believe me. Anyway, that's all the time I have in this video. In the next video, I'll introduce you to the last two logarithm properties. And if we have time,... | Introduction to logarithm properties Logarithms Algebra II Khan Academy.mp3 |
Or another way to think about it is you can distribute this negative sign along all of those terms. That's essentially what we're about to do here. We're just adding the negative of this entire thing. We're adding the opposite of it. So this first part, I'm not going to change it. That's still just 16x plus 14. But now... | Example 3 Subtracting polynomials Algebra I Khan Academy.mp3 |
We're adding the opposite of it. So this first part, I'm not going to change it. That's still just 16x plus 14. But now I'm going to distribute the negative sign here. So negative 1 times 3x squared is negative 3x squared. Negative 1 times positive x is negative x, because that's positive 1x. Negative 1 times negative ... | Example 3 Subtracting polynomials Algebra I Khan Academy.mp3 |
But now I'm going to distribute the negative sign here. So negative 1 times 3x squared is negative 3x squared. Negative 1 times positive x is negative x, because that's positive 1x. Negative 1 times negative 9, remember you have to consider this negative right over there. That is part of the term. Negative 1 times nega... | Example 3 Subtracting polynomials Algebra I Khan Academy.mp3 |
Negative 1 times negative 9, remember you have to consider this negative right over there. That is part of the term. Negative 1 times negative 9 is positive 9. Negative times a negative is a positive. So then we have positive 9. And now we just have to combine like terms. So what's our highest degree term here? | Example 3 Subtracting polynomials Algebra I Khan Academy.mp3 |
Negative times a negative is a positive. So then we have positive 9. And now we just have to combine like terms. So what's our highest degree term here? I like to write it in that order. We have only one x squared term, second degree term. We only have one of those, so let me write it over here. | Example 3 Subtracting polynomials Algebra I Khan Academy.mp3 |
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