WEBVTT mathematics/algebra-2/eaton
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Welcome to Educator.com.
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Today, we are going to start talking about polynomial functions; and we are going to begin with some review of properties of exponents.
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So, recall that, when you need to simplify an exponential expression, you need to write the expression without parentheses or negative exponents.
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Also recall that the monomial is in simplest form if each base appears only once,
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if there are no powers of powers, and all fractions are simplified.
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And recall that, in a polynomial, the terms are monomials.
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OK, first reviewing the concept of negative exponents: if a does not equal 0
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(and we have that limitation because we don't want to have a 0 in the denominator),
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a to a negative power equals 1 over a to that power.
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So, a^-n equals 1/a^n: for example, if I had y^-4, I could rewrite that as 1/y⁴.
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Now, let's look and think about why this would be.
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If you have something like x³/x⁵, that would actually give you x times x times x, all over x times x, and on 5 times.
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So, if I go ahead and look at what that would be, I could cancel out these x's (the first three).
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And I would end up with 1/x times x, or 1/x².
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I could look at this another way: I could use my rules for dividing exponents of the same base.
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And those rules would tell me that what I need to do is take x^3 - 5; so I need to subtract the exponents.
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And this is going to give me x^-2; well, since these two are equal, these two must be equal.
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Therefore, 1/x² equals x^-2; and that is why we say that these two are equivalent.
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a^-n equals 1/a^n.
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And remember: in order to simplify exponential expressions, you need to make sure that there are no negative exponents.
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And using this rule is how we get rid of those negative exponents.
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OK, reviewing properties of exponents: each of these is covered in detail in Algebra I--this is just a brief review.
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The first property is the one that we just talked about, which is negative exponents: a^-n = 1/a^n.
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The second property is a review of multiplying, where you have exponents with the same base.
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So, if you have exponents with the same base, such as x³, and you are asked
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to multiply that times something like x⁴, you accomplish that by simply adding the exponents to get x⁷.
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Division: to divide exponential expressions with the same base (I have two monomials here with the same base),
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what I am going to do is subtract the exponents, again with the limitation
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that a does not equal 0, because we cannot have 0 in the denominator, since that would be undefined.
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So, if I had something such as y⁴/y⁶, this is going to give me y^4 - 6, or y^-2.
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And today, again, I am reviewing these; but we are also going to go on and apply the properties more to negative exponents than we did in Algebra I.
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And you could further simplify this by writing it as 1/y².
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Now, raising a power to a power: when you raise a power to a power, you are going to do that by multiplying the exponents.
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So, if I have x⁴ raised to the second power, I am going to rewrite this as x^4 x 2, or x⁸.
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And again, this works with negative exponents as well.
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If I have z³, raised to the -2 power, it is going to give me z^3 x -2, or z^-6.
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So, this works for negative exponents.
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Parentheses: recall that, in order to simplify exponential expressions, we need to get rid of parentheses.
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So, if you have something such as ab, and that whole expression is raised to the m power, you can rewrite this as a^m times b^m.
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So, if I have something such as 3x, and it is squared, I can rewrite this as 3²x², which would give me 9x².
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Again, parentheses, but now talking about division: if I have a divided by b,
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all raised to the n power, I can rewrite that as a^n divided by b^n,
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with the restriction that b, since it is in the denominator, cannot equal 0.
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For example, if I have x/y raised to the fourth power, I could rewrite that as x⁴ divided by y⁴.
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Finally, something raised to a zero power; if I have a raised to a 0 power, it equals 1.
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However, we again have the restriction where a cannot equal 0, because this is not defined.
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So, we don't work with that.
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Now, let's think a little bit more deeply about why this is true.
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We are saying that a⁰ equals 1; why is that so?
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Well, it is simply because we defined it that way, to make everything work out, and all the rules be consistent.
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And you can look at it this way: let's say I have y³ over y³.
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Well, using my rule of division that I just talked about, when I have exponents with like bases,
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I could say, "OK, this is y^3 - 3, or y⁰."
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Well, I could also look at it another way: y³/y³ equals y times y times y, all over y times y times y.
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OK, so now these cancel out; and what this gives me is 1/1, or 1.
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Now, to make everything work out and have everything (all these rules) be consistent, what I am saying
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is that these two are equal; and I used a rule here and a rule here, and I got these two different things;
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therefore, these two things must be equal, in order to be able to use all of these rules.
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So, we say that any number to the 0 power is 1.
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Now, applying these rules to simplify some exponential expressions:
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recall that, in order to be in simplest form, I need to make sure there are no negative exponents (which there are not),
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no parentheses (I have parentheses), and no powers raised to powers (I have that, so I need to take care of all that).
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So, I am going to start out by recalling that a power raised to a power is equal to a^m x n.
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So, let's first get rid of these parentheses by saying that I actually have
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(a²)⁴ times (b³)⁴ times (c²)⁴
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times (b²)³ times (c³)³ times a³.
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Now, all I need to do is multiply these out; and this is going to give me...I have powers raised to powers,
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so a⁸ times b^12 times c⁸ times...2 times 3 is 6, so that is b⁶,
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times c⁹, times a³; so, this monomial is still not in simplest form,
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because I still have some bases that are duplicate here.
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Each base should be represented only once in the monomial.
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So, in order to simplify this, I need to multiply.
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And what I can do is recall my rule for multiplication when exponents have the same base.
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a^m times a^n equals a^m + n.
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So, what I can do is say, "OK, I have a to the eighth power, and I have a to the third power, so I am going to add those exponents."
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I have b to the twelfth power, and I have b to the sixth power; so I am going to add those.
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I have c to the eighth, and I have c to the ninth; so I am going to add those.
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And this is going to give me a^11b^18c^17.
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So, this is now in my simplest form; and I accomplished that by getting rid of the parentheses and using my rule for raising the power to a power.
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So, I multiplied each of these times its power; and then I found that I had bases represented more than once.
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So, I multiplied the expressions that had like bases by adding the exponents.
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OK, simplify: here we have parentheses; we have powers to powers; and we have negative exponents.
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So, I need to get rid of all that, first by applying the power to each term inside the parentheses.
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This is going to give me 2⁸ times (x²)⁸ times (y³)⁸
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times (z⁴)⁸ times x^-2y^-2(z²)^-4.
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OK, simplifying: if you work this out to multiply 2 by itself 8 times,
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you would find that you are going to get 256 for that one.
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Now, here I have 2 times 8; that is 16; 3 times 8 is 24; z^4 x 8--that is 32.
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And then here, I have a bunch of negative exponents: x^-2y^-2...z^2 x -4 is z^-8.
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Now, recall that, if I have a^-n, this equals 1/a^n.
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So, in order to eliminate these negative exponents, I am going to move all three of these into the denominator.
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But this is actually an error right here; OK, let me correct that.
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This is z squared to the negative 2, so this would be 2 times -2; that is actually negative 4; so this is z to the fourth power right here.
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Now, I can further simplify by dividing; recall that dividing exponential expressions with like bases, you subtract the exponents.
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I can do that here: this is going to give me 256x^16 - 2, times y^24 -2, times z^32 - 4.
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Now, I do my subtraction to get x raised to the fourteenth power, y to the twenty-second power, and z to the twenty-eighth.
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And this is my expression in simplest form; and I can verify that, because each base is represented only once;
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there are no powers to powers; and there are no negative exponents; and there are no parentheses.
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Now, this was kind of complicated; but I started out by applying the exponents to each of the numbers and variables inside the parentheses.
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And I multiplied using my rule for a power to power, which is (a^m)^n = a^mn.
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OK, once I did that, I saw negative exponents; and I used this rule to move those into the denominator, so they became positive.
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And then, I had like bases, so I used my rule for division, in order to further simplify, getting this as the final result.
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OK, simplify: here I have a negative power, and I have items inside of parentheses raised to that negative power.
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Recall this rule: if I have a/b, all raised to a certain power, this equals a to that power, over b to that power.
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So, what I can do is apply this -4 to the numerator and to the denominator, separately.
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So, this is going to give me a³b²x⁴, all to the negative fourth power,
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over x²y³c³, all to the negative fourth.
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OK, once I have that, I am going to use my power to a power rule: (a^m) raised to the n power equals a^mn.
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OK, so I am applying the -4 to each item inside the parentheses and doing the same thing in the denominator,
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and then, multiplying the exponents to get a^-12b...2 times -4 gives me -8;
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and then here, I have 4 times -4 is going to give me -16: x^-16; x^-8y^-12c^-12.
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Before I go any farther and deal with the negative exponents,
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I first notice that I have some simplifying I can do, because these two have the same base.
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Therefore, I can divide; and dividing will require me to subtract the exponents (dividing with like bases).
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a^-12b^-8x^-16 -...and this is a negative, so it is - -8.
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And now, in the bottom, I just have y^-12c^-12.
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OK, this is going to give me a^-12b^-8...-16, and this is minus - -8, so it is really + 8;
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-16 plus 8 is going to leave me with -8; in the denominator, I have y^-12c^-12.
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Now, I have a lot of these negative exponents.
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Recall the rule for negative exponents: a^-n = 1/a^n.
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Therefore, if I move everything in the numerator to the denominator, those exponents will become positive.
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For the ones in the denominator, I have to switch those to the numerator for them to become positive.
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Therefore, I will end up with y^12c^12 over a^12b⁸x⁸.
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OK, so this is simplest form, because I have no parentheses; I have no powers to powers; and I have no negative exponents.
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I started out by splitting the numerator and the denominator by applying this power separately to each.
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Then, I applied the power to each item inside the parentheses to come up with this.
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Then, I noticed that I had the x's, and they had the same base, so I divided x^-16 by x^-8 to get this.
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And finally, I was eliminating these negative powers by using this rule
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and switching the items in the numerator and the denominator, to get my simplest form.
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OK, we are asked to simplify; and we have bases that are represented twice, and we also have negative exponents that we need to get rid of.
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First, I am going to start out by dividing; I have a^-3, and then that is going to be,
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using my rule for division, a^m divided by a^n equals a^m - n...
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-3 minus -4; b^2 - -3...you have to be careful with the signs with all these negatives...
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and then c^-2 - -4; let's take care of these signs.
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This is -3; a negative and a negative actually gives me + 4.
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And this is 2, minus -3, so that is + 3; and then c, minus -4, is + 4.
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So, this gives me -3 + 4 (is just 1, so I leave that as a); b: 2 + 3 is 5; and then c: -2 + 4 is 2.
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So, this started out looking very messy; but actually, once you divided, it took care of all of those negative exponents.
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That concludes this lesson of Educator.com on exponential expressions.
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And I will see you soon!