Algebra 2 Properties of Exponents

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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.0002

So, recall that, when you need to simplify an exponential expression, you need to write the expression without parentheses or negative exponents.0009

Also recall that the monomial is in simplest form if each base appears only once,0020

if there are no powers of powers, and all fractions are simplified.0041

And recall that, in a polynomial, the terms are monomials.0055

OK, first reviewing the concept of negative exponents: if a does not equal 00067

(and we have that limitation because we don't want to have a 0 in the denominator),0073

a to a negative power equals 1 over a to that power.0077

So, a^-n equals 1/aⁿ: for example, if I had y^-4, I could rewrite that as 1/y⁴.0082

Now, let's look and think about why this would be.0103

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.0107

So, if I go ahead and look at what that would be, I could cancel out these x's (the first three).0122

And I would end up with 1/x times x, or 1/x².0130

I could look at this another way: I could use my rules for dividing exponents of the same base.0137

And those rules would tell me that what I need to do is take x^{3 - 5}; so I need to subtract the exponents.0142

And this is going to give me x^-2; well, since these two are equal, these two must be equal.0154

Therefore, 1/x² equals x^-2; and that is why we say that these two are equivalent.0161

a^-n equals 1/aⁿ.0169

And remember: in order to simplify exponential expressions, you need to make sure that there are no negative exponents.0172

And using this rule is how we get rid of those negative exponents.0179

OK, reviewing properties of exponents: each of these is covered in detail in Algebra I--this is just a brief review.0186

The first property is the one that we just talked about, which is negative exponents: a^-n = 1/aⁿ.0194

The second property is a review of multiplying, where you have exponents with the same base.0204

So, if you have exponents with the same base, such as x³, and you are asked0211

to multiply that times something like x⁴, you accomplish that by simply adding the exponents to get x⁷.0216

Division: to divide exponential expressions with the same base (I have two monomials here with the same base),0225

what I am going to do is subtract the exponents, again with the limitation0233

that a does not equal 0, because we cannot have 0 in the denominator, since that would be undefined.0237

So, if I had something such as y⁴/y⁶, this is going to give me y^{4 - 6}, or y^-2.0244

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.0256

And you could further simplify this by writing it as 1/y².0268

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.0272

So, if I have x⁴ raised to the second power, I am going to rewrite this as x^{4 x 2}, or x⁸.0280

And again, this works with negative exponents as well.0292

If I have z³, raised to the -2 power, it is going to give me z^{3 x -2}, or z^-6.0296

So, this works for negative exponents.0308

Parentheses: recall that, in order to simplify exponential expressions, we need to get rid of parentheses.0312

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.0318

So, if I have something such as 3x, and it is squared, I can rewrite this as 3²x², which would give me 9x².0331

Again, parentheses, but now talking about division: if I have a divided by b,0348

all raised to the n power, I can rewrite that as aⁿ divided by bⁿ,0354

with the restriction that b, since it is in the denominator, cannot equal 0.0358

For example, if I have x/y raised to the fourth power, I could rewrite that as x⁴ divided by y⁴.0362

Finally, something raised to a zero power; if I have a raised to a 0 power, it equals 1.0376

However, we again have the restriction where a cannot equal 0, because this is not defined.0384

So, we don't work with that.0391

Now, let's think a little bit more deeply about why this is true.0394

We are saying that a⁰ equals 1; why is that so?0397

Well, it is simply because we defined it that way, to make everything work out, and all the rules be consistent.0401

And you can look at it this way: let's say I have y³ over y³.0408

Well, using my rule of division that I just talked about, when I have exponents with like bases,0414

I could say, "OK, this is y^{3 - 3}, or y⁰."0421

Well, I could also look at it another way: y³/y³ equals y times y times y, all over y times y times y.0427

OK, so now these cancel out; and what this gives me is 1/1, or 1.0438

Now, to make everything work out and have everything (all these rules) be consistent, what I am saying0446

is that these two are equal; and I used a rule here and a rule here, and I got these two different things;0452

therefore, these two things must be equal, in order to be able to use all of these rules.0459

So, we say that any number to the 0 power is 1.0467

Now, applying these rules to simplify some exponential expressions:0480

recall that, in order to be in simplest form, I need to make sure there are no negative exponents (which there are not),0484

no parentheses (I have parentheses), and no powers raised to powers (I have that, so I need to take care of all that).0490

So, I am going to start out by recalling that a power raised to a power is equal to a^{m x n}.0497

So, let's first get rid of these parentheses by saying that I actually have0510

(a²)⁴ times (b³)⁴ times (c²)⁴0515

times (b²)³ times (c³)³ times a³.0527

Now, all I need to do is multiply these out; and this is going to give me...I have powers raised to powers,0537

so a⁸ times b¹² times c⁸ times...2 times 3 is 6, so that is b⁶,0547

times c⁹, times a³; so, this monomial is still not in simplest form,0558

because I still have some bases that are duplicate here.0566

Each base should be represented only once in the monomial.0570

So, in order to simplify this, I need to multiply.0573

And what I can do is recall my rule for multiplication when exponents have the same base.0577

a^m times aⁿ equals a^{m + n}.0584

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."0590

I have b to the twelfth power, and I have b to the sixth power; so I am going to add those.0597

I have c to the eighth, and I have c to the ninth; so I am going to add those.0603

And this is going to give me a¹¹b¹⁸c¹⁷.0608

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.0616

So, I multiplied each of these times its power; and then I found that I had bases represented more than once.0626

So, I multiplied the expressions that had like bases by adding the exponents.0633

OK, simplify: here we have parentheses; we have powers to powers; and we have negative exponents.0641

So, I need to get rid of all that, first by applying the power to each term inside the parentheses.0648

This is going to give me 2⁸ times (x²)⁸ times (y³)⁸0654

times (z⁴)⁸ times x^-2y^-2(z²)^-4.0664

OK, simplifying: if you work this out to multiply 2 by itself 8 times,0674

you would find that you are going to get 256 for that one.0681

Now, here I have 2 times 8; that is 16; 3 times 8 is 24; z^{4 x 8}--that is 32.0687

And then here, I have a bunch of negative exponents: x^-2y^-2...z^{2 x -4} is z^-8.0707

Now, recall that, if I have a^-n, this equals 1/aⁿ.0716

So, in order to eliminate these negative exponents, I am going to move all three of these into the denominator.0722

But this is actually an error right here; OK, let me correct that.0745

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.0748

Now, I can further simplify by dividing; recall that dividing exponential expressions with like bases, you subtract the exponents.0757

I can do that here: this is going to give me 256x^{16 - 2}, times y^{24 -2}, times z^{32 - 4}.0768

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.0788

And this is my expression in simplest form; and I can verify that, because each base is represented only once;0799

there are no powers to powers; and there are no negative exponents; and there are no parentheses.0806

Now, this was kind of complicated; but I started out by applying the exponents to each of the numbers and variables inside the parentheses.0811

And I multiplied using my rule for a power to power, which is (a^m)ⁿ = a^mn.0823

OK, once I did that, I saw negative exponents; and I used this rule to move those into the denominator, so they became positive.0834

And then, I had like bases, so I used my rule for division, in order to further simplify, getting this as the final result.0841

OK, simplify: here I have a negative power, and I have items inside of parentheses raised to that negative power.0851

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.0860

So, what I can do is apply this -4 to the numerator and to the denominator, separately.0870

So, this is going to give me a³b²x⁴, all to the negative fourth power,0876

over x²y³c³, all to the negative fourth.0882

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.0889

OK, so I am applying the -4 to each item inside the parentheses and doing the same thing in the denominator,0899

and then, multiplying the exponents to get a^-12b...2 times -4 gives me -8;0925

and then here, I have 4 times -4 is going to give me -16: x^-16; x^-8y^-12c^-12.0936

Before I go any farther and deal with the negative exponents,0954

I first notice that I have some simplifying I can do, because these two have the same base.0957

Therefore, I can divide; and dividing will require me to subtract the exponents (dividing with like bases).0962

a^-12b^-8x^{-16 -}...and this is a negative, so it is - -8.0971

And now, in the bottom, I just have y^-12c^-12.0982

OK, this is going to give me a^-12b^-8...-16, and this is minus - -8, so it is really + 8;0986

-16 plus 8 is going to leave me with -8; in the denominator, I have y^-12c^-12.0997

Now, I have a lot of these negative exponents.1007

Recall the rule for negative exponents: a^-n = 1/aⁿ.1010

Therefore, if I move everything in the numerator to the denominator, those exponents will become positive.1015

For the ones in the denominator, I have to switch those to the numerator for them to become positive.1022

Therefore, I will end up with y¹²c¹² over a¹²b⁸x⁸.1027

OK, so this is simplest form, because I have no parentheses; I have no powers to powers; and I have no negative exponents.1038

I started out by splitting the numerator and the denominator by applying this power separately to each.1046

Then, I applied the power to each item inside the parentheses to come up with this.1052

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.1060

And finally, I was eliminating these negative powers by using this rule1070

and switching the items in the numerator and the denominator, to get my simplest form.1076

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.1083

First, I am going to start out by dividing; I have a^-3, and then that is going to be,1094

using my rule for division, a^m divided by aⁿ equals a^{m - n}...1101

-3 minus -4; b^{2 - -3}...you have to be careful with the signs with all these negatives...1108

and then c^{-2 - -4}; let's take care of these signs.1121

This is -3; a negative and a negative actually gives me + 4.1128

And this is 2, minus -3, so that is + 3; and then c, minus -4, is + 4.1133

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 2.1143

So, this started out looking very messy; but actually, once you divided, it took care of all of those negative exponents.1153

That concludes this lesson of Educator.com on exponential expressions.1163

And I will see you soon!1168