Algebra 2 Rational Exponents

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Today, we will be talking about rational exponents.0002

And this is a concept that may be new, so let's start out by talking about what rational exponents are.0005

They are actually exponents that are fractions; so you may also hear these called fractional exponents.0012

OK, starting out with a definition: for a positive integer n, a^1/n = the n^th root of a.0020

We will talk about this restriction in a second: first, let's just look at what this is saying, using numbers.0034

So, for a positive integer n, a^1/n equals this.0040

So, let's let a equal 5 and n equal 2; if a equals 5 and n equals 2, then this is going to give me 5^1/2, to the one-half power.0047

This equals the square root of 5, since we don't actually write the 2.0065

When you see this notation, what you need to do is take this number to the whatever root this denominator is.0072

This...we have talked before that n, right here by this radical, is the index.0084

So, in this case, we put the index right here: 1 over the index; it is the number to 1 over the index.0091

You may also see this with other values of n, and with variables involved.0100

3x to the 1/3 power equals...well, this is the index, so I am going to put it right here...the cube root of this 3x.0106

I could proceed in the other direction: I may have something that is saying "the sixth root of a."0116

And I may be asked to write it using a rational exponent or a fractional exponent--an exponent that is a fraction.0124

OK, so I write the a here; and I do 1 over the index; the index here is 6, n = 6, so this is 8^1/6.0132

Now, let's look at this restriction: and we have seen similar restrictions when we have even indices in other situations.0144

If n is even, then a must be greater than or equal to 0.0152

a must be a non-negative number when n is even: let's think about why.0159

If I have something such as -4, and n is 2; so I am letting a equal -4 and n equal 2; that would be written as such.0166

Well, this equals the square root of -4, and my index is 2; so it is just written as a square root.0181

This is going to end up giving me something that is not a real number; and we are just staying within real numbers,0190

restricting our discussion to real numbers, although we have learned about imaginary and complex numbers in this course.0195

Therefore, we are not allowing that; we are going to have this restriction that, if n is even, then a must be greater than or equal to 0.0201

I don't run into that problem if n is odd; and the reason is--let's say n is 3, and I am going to have a equal -8.0210

a is allowed to equal a negative number if n is odd.0221

This is going to be -8^1/3; -8 goes here, and this is the index; so it is the cube root of -8.0225

Recall that -2 cubed would be -2 times -2 times -2, which would be -8.0237

Therefore, the cube root of this negative number is a real number, whereas an even root of a negative number is not going to give you a real number.0252

That is why this restriction is only for when n is even.0264

A more general case: for positive integers m and n, a^m/n equals the n root of a^m.0272

Or we can also rewrite this as the n^th root of a, raised to the m power.0282

Again, we have the restriction that, if n is even, then a must be greater than or equal to 0.0292

In the last case we saw, that was just a special case of this where m was 1.0298

So, before, we said that, if we had something like 4^1/2, I could just say, "OK, m equals 1, a equals 4, and n equals 2."0304

And then, I am rewriting that as the square root (because the index is 2) of 4.0317

And m was 1, so I didn't actually write it; I don't write all this out in this case--I just say the square root of 4.0322

Now, we are talking about situations where m is something other than 1, but it is the same basic concept.0331

For example, if I have 2^3/4, here m equals 3; a equals 2; and n equals 4.0338

So then, I am going to have 2 under the radical; this is going to be the fourth root of 2³.0350

With variables, you could have something like x^3/2, and that is going to give me x under the radical;0360

the index is 2; and the power that x is raised to is 3.0370

Again, we have this restriction that if n is even, right here, then a must be greater than or equal to 0.0376

So, even if m is something other...for example, m could be 7/2...this would be...3 goes into the radical;0386

this is the index, n, and then m is to the seventh power.0411

So, since this is even, I do have the restriction that whatever is under here has to be positive.0414

So, if I had a variable, this would give me x¹³; that is the fourth root of x to the thirteenth power.0419

I have the restriction here that x has to be greater than or equal to 0.0435

OK, so again, you look at what you have; you put this number under the radical.0444

The denominator is the index; the numerator is the power that the radicand is raised to.0451

And for even indexes, you need to have the restriction that the radicand must be greater than or equal to 0.0458

We talked, in the last lecture, about simplifying radical expressions.0474

And we are going to talk in more depth now; and we are going to start out by just reviewing some properties0480

that we talked about for when we are working with powers.0487

And in earlier lessons, in Algebra I, you learned these properties; and we are going to review them now,0491

because we are going to apply them to the situation where we have rational exponents.0498

So, you learn these properties for when the exponents are integers; but they still hold up when you have fractions as the exponents.0503

For example, multiplication: a^m times aⁿ equals a^{m + n}.0512

And we did that working with things like x³ times x².0525

But this also applies where we might have something such as x^1/2 times x^3/4.0532

So, this is just going to equal x to the 1/2 + 3/4; so adding the exponents, this is going to be just 2/4 + 3/4, so this is going to be x^5/4.0540

It is the same property, just using fractions up here for m and n.0562

OK, when you are working with quotients (with division), a^m/aⁿ equals a, and then you subtract:0567

You take m and subtract the exponent in the denominator; again, this holds true with division, if you have something with a rational exponent.0579

So, if you have something like y^2/3, divided by y^1/3, I just do the same thing.0589

y^{2/3 - 1/3} = y^1/3.0601

Recall that a^m, raised to the n power, is a^{m x n}; you multiply the exponents.0609

The same thing here--we might have something like wz to the 1/6^th is the same as w^1/6 times z^1/6.0618

Oh, excuse me, that is the next one.0633

You might have something such as x^1/2 raised to the 3/4 power, which is going to give you x to the 1/2 times 3/4.0639

And this is going to equal x to the 3 over 2(4), so that is x^3/8.0655

Now, when you have something like ab, and all of that is raised to a power, that gave you a^m times b^m.0664

Here, what I started to show you before was (wz)^1/6 = w^1/6z^1/6.0676

Now, another property to recall is that a (any number) to the 0 power equals 1.0688

But recall that a does not equal 0, because 0 to the 0 power is not defined.0699

And finally, something that we are going to be using again to simplify, which we used0709

in earlier lessons, is that a^-n = 1/aⁿ; and this also applies when you are working with fractions.0712

So, if I had something such as a^-1/5, this equals 1/a^1/5.0721

Now, this first states that the properties of powers for integer exponents that we learned are valid for rational exponents.0735

And that is what I demonstrated here.0743

A couple more rules about simplifying: simplified expressions--we talked earlier on about what a simplified expression looks like.0746

when working with radicals--that we can't have radicals in the denominator; that we cannot have fraction beneath the radical...0757

Well, in addition, you will recall that simplified expressions contain only positive exponents,0766

and that exponents in the denominator must be positive integers.0772

So, if you have something like xy to the -2, that is not simplified.0778

Also, if I have something like 2x/y^1/2, this is also not simplified.0789

So, wherever the exponents are (numerator or denominator), they need to be positive in order to have something be in simplest form.0798

For the denominator, in addition to saying that all of the exponents must be only positive,0807

we also say that we cannot have fractional (rational) exponents in the denominator.0813

Exponents in the denominator must be integers; so these are not simplified.0819

Let's talk now about simplifying and sum up what we have learned so far.0830

An expression with rational exponents is simplified if it has no negative exponents, there are no rational0836

or fractional exponents in the denominator, and it is not a complex fraction.0843

We just covered these two a bit; just recalling what we are talking about, you can't have a complex fraction0848

So, if you have something like 2 divided by x, over y⁴, divided by 4, a complex fraction--that is obviously not in simplest form.0856

And the index needs to be as small as possible.0867

Again, we are not going to be working with this too much; but it is just something to be aware of.0871

Let's focus on these two: first, not having negative exponents.0875

Since a^-n equals 1/aⁿ, we can use this property in order to get rid of negative exponents.0880

So, if I have something such as 2x^-3, I can simplify this by saying,0888

"OK, I will put x in the denominator, and then I can change this number to a positive."0893

Because a^-1 equals 1/aⁿ, that means that x^-3 is the same as 1/x³.0904

Let's look at a little bit more complicated situation, though.0915

Let's look at x^-3/4: well, looking at this property, I see how to get rid of the negative--that I just rewrite it as 1/x^3/4.0918

So, I am looking through, and I said, "OK, I have no negative exponents; great!"0934

But then, I look at this next rule: an expression is simplified if it has no fractional exponents in the denominator.0937

So, even though now I have gotten rid of that negative exponent, I have created a new problem.0945

And that is that I have a fractional exponent in the denominator.0950

So, let's think about how to get rid of that.0954

The way we are going to get rid of that is: I want to change this 3/4 to 1.0956

That way, this will just become x--let's think about how I can do that.0961

x (or I am going to write it out as x¹ in this case) equals x^3/40967

times x to something--we are just going to call that other exponent y.0976

Well, since multiplication involves adding exponents, what I am really saying is this: x¹ = x^{3/4 + y}.0982

So, I need to figure out what y is; and I know, in this one, it is pretty simple just from looking at it.0996

But this is a technique you can use with more complex problems.1000

1 = 3/4 + y; solving for y, y = 1/4; so what this tells me is that x¹ = x^3/4 times x^1/4,1003

which you probably just figured out from looking; but again, this is a technique that you might need to use.1023

Therefore, I know that what I can do is multiply this times x^1/4.1029

And if I do, it is going to give me an x in the denominator.1043

However, I need to also multiply the numerator by x^1/4, because that way, this divided by this is just 1.1047

So I am really just multiplying by 1.1055

So, this is going to give me x^1/4 over x; now, I have no negative exponents;1058

I have no fractional exponents in the denominator; and this is not a complex fraction; and the index is as small as possible.1068

I could even...the index here is 4; something else I could do, just to check1077

and make sure that what I ended up with is equivalent to what I started with--1083

I could use my quotient property and say, "OK, this is equal to x^{1/4 - 1}."1087

Division would be x^1/4 minus the power down here--x to the 1/4 minus 1 equals x to the -3/4.1102

So, I see that I got this back; therefore, this is equivalent to what I started out with, but it is written in simplified form, because it meets these criteria.1114

So again, the main new technique to learn (because this is review) is that,1125

if you have a fractional exponent in the denominator, you want to multiply by something1129

that will make this exponent 1, to get rid of that fractional exponent.1137

So, let's work out some examples.1144

Example 1 asks us to write this expression as a radical: (8c²)^3/5.1147

Recall that a^m/n = ⁿ√a^m.1155

Here, a equals 8c²; it looks complicated, but it is just my a; m is 3, and n is 5.1167

So, right here, I am going to have 8c² under the radical; m is 3, so I am going to raise this to 3.1181

The index is 5, so I am going to take the fifth power of that.1193

I could leave it like this, or I could go one step further and say, "OK, if you worked out 8 to the fifth power, you would get that it is 512."1197

And this is c squared to the third power; recall that, if I am going to raise a power to a power, I am going to multiply the exponents.1207

So, it is c⁶; now this is in the simplest form that I can get it in, because this is equivalent to 8³ times (c²)³.1221

We were using this rule that shows us how to convert a fractional exponent into a radical.1231

Example 2: now we are given a radical and asked to write it with rational exponents.1241

So again, I am going to think about my definition that a^m/n = a...index is n, and it is raised to the m power.1247

So here, what I have is a = 3x² - 2; m is right here--m is 4; and n is 5.1261

With that in mind, I can rewrite this as 3x...actually, we are working the other way, so I am going to rewrite this as--1273

no radical--3x² - 2, and here I have m, so m becomes the numerator, and that is 4; and the denominator is 5.1287

So, this expression, the fifth root of 3x² - 2 to the fourth, equals 3x² - 2 to the 4/5.1307

All I needed to do is eliminate the radical, and then raise this to the 4/5 power.1319

And these are equivalent expressions.1326

Example 3: I am asked to simplify, and I am going to think about my rules of simplifying.1330

It is that an expression is in simplest form if it has no negative exponents;1335

if it has no fractional exponents in the denominator; if it is not a complex fraction;1345

and if the index is as small as possible.1376

So, I see here that I don't have any negative exponents right now, but I do have some fractional exponents.1380

So, what I am working with is division; so I am just going to start out by dividing.1388

This x^1/2 divided by x^2/3 follows the property of exponents, when we are working with quotients,1393

that tells me that I take x to the 1/2 - 2/3.1400

And this equals x to the...we are looking for the common denominator of 6, so this is x to the 3/6 - x to the 4/6.1407

So, this equals x^-1/6.1422

Now, I got rid of the fractional exponent in the denominator; but now what I have is a negative exponent, x^-1/6.1427

To get rid of that, I am going to use my rule that tells me that, if I have a negative exponent, I can convert it to this: 1/x^1/6.1438

Now, I have something simpler-looking that I can work with.1452

I am back to having this fractional exponent in the denominator, but this is simpler to work with.1454

OK, so recall that the way we get rid of a fractional exponent is: we want to convert this to the exponent of 1 instead.1460

So, I want x¹, and I have x^1/6, so I need to multiply this by x to something,1468

which is the same as x^{1/6 + y}.1477

So, I am going to set 1 equal to 1/6 + y; this gives me 1 - 1/6 = y, so y equals 5/6.1484

So, what this tells me is that I need to multiply this times x^1/6.1494

Only, I have to do the same thing to the numerator and the denominator, so I am going to also multiply1507

the numerator by x^5/6, so that really, I am just multiplying by 1, which is allowable.1513

And this is going to give me x^5/6, and x^{1/6 + 5/6}; that is just x.1519

So, I end up with x^5/6/x; this may not look that much simpler than what we started out with,1531

but it meets the criteria for simplest form, because I now have no negative exponents;1537

I have no fractional exponents in the denominator; and this is not a complex fraction.1542

So, this is in simplest form.1548

I started out by just using my regular rule for dividing with exponents, which means that I am going to subtract the exponents.1550

Since there are like bases, I just subtract the exponents.1557

I have -1/6; I use my rule that tells me I can rewrite this as 1/x^1/6 to get rid of the negative.1560

And then, I used the technique of multiplying the numerator and the denominator by x^5/6,1569

because that will make this x; and I am not worried about having a fractional exponent in the numerator, because that is allowed.1575

In this fourth example, we have x^5/2 over x^1/2 - 2.1582

Now, recall what x^1/2 is: it is actually just the square root of x.1588

So, what we have here is really the square root of x, minus 2.1594

And recall that, if there is a radical in the denominator, or a fractional exponent in the denominator,1599

which is really the same thing, then it is not in simplest form.1605

And in an earlier lesson, we talked about how, if you have a radical in the denominator that is part of a binomial,1608

you multiply both the numerator and the denominator by the conjugate, in order to eliminate that radical.1614

So, it is the same idea here, only we are working with it in this form: x^1/2 - 2.1622

The conjugate is going to be x^1/2 + 2, which is the same as up here, if I were to say that the conjugate would be √x + 2--same idea.1630

I need to multiply both the numerator and the denominator by that.1649

And the reason that I have to multiply both the numerator and the denominator is because,1660

that way, I am really just multiplying by 1, which is allowable.1664

This is really just multiplying by 1.1670

Using the distributive property in the top: x^5/2, times this first term, x^1/2, plus x^5/2, times 2.1672

In the denominator, what we can do is use the fact that, when we are multiplying conjugates like this,1690

x^1/2 - 2 and x^1/2 + 2, it is multiplying a sum and a difference.1697

So, it is in this form: here we have the difference first, so I will write it that way.1705

But you will end up with something in this form: a² - b².1709

In this case, here, a is equal to x^1/2, and b is equal to 2.1713

Therefore, what I am going to end up with is (x^1/2)² - 2².1720

So, I am going to put that right down here: (x^1/2)² - 2².1727

OK, since multiplying with like bases, when you have an exponent, just means to add the exponents; I am going to do the same thing here.1736

This really is just x^{5/2 + 1/2}; so, 5/2 + 1/2 is just going to give me 6/2.1745

Plus...I am going to rewrite this as 2 times x^5/2.1758

I go to the denominator, and that is going to give me (x^1/2)².1765

Well, using the rules of raising a power to a power, this is going to be x^{1/2 x 2}, which is going to be x^2/2, or x¹.1771

So, it is just x; 2 squared is 4; this is going to give me 6 divided by 2, which is x³, plus 2x^5/2, divided by x - 4.1788

So, it is still a pretty complicated-looking expression.1806

However, when I look, it is in simplified form.1810

I no longer have a fractional exponent in the denominator; and in order to get rid of that, it is the same concept1813

as getting rid of a radical in the denominator that is part of a binomial, because that is actually what I really have.1820

It is just written with a different notation.1826

I multiplied the numerator and the denominator by the conjugate.1828

That got rid of this fractional exponent in the denominator, and then it just came down to doing some simplifying.1832

That concludes this session of Educator.com; thanks for visiting!1841