### Rational Exponents

- All the properties of integer valued exponents remain true for rational exponents.
- In simplified form, all exponents must be positive and exponents in the denominator must be integers.

### Rational Exponents

^{3})

^{[4/7]}

- Recall the formula a
^{[m/n]}=^{n}√{a^{m}} - Identify
- a =
- m =
- n =
- a = 9x
^{3} - m = 4
- n = 7
- Plug - in the values
- (9x
^{3})^{[4/7]}=^{n}√{a^{m}} =^{7}√{( 9x^{3})^{4}} - Simplify

^{7}√{( 9x

^{3})

^{4}} =

^{7}√{6561x

^{12}}

^{4})

^{[2/3]}

- Recall the formula a
^{[m/n]}=^{n}√{a^{m}} - Identify
- a =
- m =
- n =
- a = 2x
^{4} - m = 2
- n = 3
- Plug - in the values
- (2x
^{4})^{[2/3]}=^{n}√{a^{m}} =^{3}√{( 2x^{4})^{2}} - Simplify

^{3}√{( 2x

^{4})

^{2}} =

^{3}√{4x

^{8}}

^{7})

^{[4/5]}

- Recall the formula a
^{[m/n]}=^{n}√{a^{m}} - Identify
- a =
- m =
- n =
- a = 5x
^{7} - m = 4
- n = 5
- Plug - in the values
- (5x
^{7})^{[4/5]}=^{n}√{a^{m}} =^{5}√{( 5x^{7})^{4}} - Simplify

^{5}√{( 5x

^{7})

^{4}} =

^{5}√{625x

^{28}}

^{5}√{(625 − x

^{2})

^{28}}

- Recall the formula a
^{[m/n]}=^{n}√{a^{m}} - Identify
- a =
- m =
- n =
- a = 625 − x
^{2} - m = 28
- n = 5
- Plug - in the values

^{5}√{(625 − x

^{2})

^{28}} = a

^{[m/n]}= (625 − x

^{2})

^{[28/5]}

^{4}√{(x

^{2}− 81)

^{3}}

- Recall the formula a
^{[m/n]}=^{n}√{a^{m}} - Identify
- a =
- m =
- n =
- a = x
^{2}− 81 - m = 3
- n = 4
- Plug - in the values

^{4}√{(x

^{2}− 81)

^{3}} = a

^{[m/n]}= (x

^{2}− 81)

^{[3/4]}

^{5}√{(x

^{2}+ x + 5)

^{2}}

- Recall the formula a
^{[m/n]}=^{n}√{a^{m}} - Identify
- a =
- m =
- n =
- a = x
^{2}+ x + 5 - m = 2
- n = 5
- Plug - in the values

^{5}√{(x

^{2}+ x + 5)

^{2}} = a

^{[m/n]}= (x

^{2}+ x + 5)

^{[2/5]}

^{[3/4]})/(x

^{[7/8]})]

- Recall that the rules of simplfying, which are:
- 1. No negative exponents
- 2. No fractional exponents in denominator
- 3. Not a complex fraction
- 4. Index as small as possible
- We are going to use these rules to guide us through the problem.
- Step 1: Using rules of exponents, simplify remember [(a
^{m})/(a^{n})] = a^{m − n} - [(x
^{[3/4]})/(x^{[7/8]})] = x^{[3/4] − [7/8]} - In order to subtract the fraction, we need to have the same denominator, in this case 8
- x
^{[3/4] − [7/8]}= x^{[6/8] − [7/8]}= x^{ − [1/8]} - Eliminate negative exponents using the rule a
^{ − n}= [1/(a^{n})] - x
^{ − [1/8]}= [1/(x^{[1/8]})] - We need to eliminate the fractional exponent in the denominator.
- We need to multiply by a number (x
^{?}) such that when multiplied with (x^{[1/8]}) equals x - Multiply numerator and denominator by (x
^{[7/8]}) - [1/(x
^{[1/8]})]*( [(x^{[7/8]})/(x^{[7/8]})] ) = [(x^{[7/8]})/x]

^{[1/5]})/(x

^{[1/2]})]

- Recall that the rules of simplfying, which are:
- 1. No negative exponents
- 2. No fractional exponents in denominator
- 3. Not a complex fraction
- 4. Index as small as possible
- We are going to use these rules to guide us through the problem.
- Step 1: Using rules of exponents, simplify remember [(a
^{m})/(a^{n})] = a^{m − n} - [(x
^{[1/5]})/(x^{[1/2]})] = x^{[1/5] − [1/2]} - In order to subtract the fraction, we need to have the same denominator, in this case 10
- x
^{[1/5] − [1/2]}= x^{[2/10] − [5/10]}= x^{ − [3/10]} - Eliminate negative exponents using the rule a
^{ − n}= [1/(a^{n})] - x
^{ − [3/10]}= [1/(x^{[3/10]})] - We need to eliminate the fractional exponent in the denominator.
- We need to multiply by a number (x
^{?}) such that when multiplied with (x^{[3/10]}) equals x - Multiply numerator and denominator by (x
^{[7/10]}) - [1/(x
^{[3/10]})]*( [(x^{[7/10]})/(x^{[7/10]})] ) = [(x^{[3/10]})/x]

^{[2/3]})/(x

^{[3/4]})]

- Recall that the rules of simplfying, which are:
- 1. No negative exponents
- 2. No fractional exponents in denominator
- 3. Not a complex fraction
- 4. Index as small as possible
- We are going to use these rules to guide us through the problem.
- Step 1: Using rules of exponents, simplify remember [(a
^{m})/(a^{n})] = a^{m − n} - [(x
^{[2/3]})/(x^{[3/4]})] = x^{[2/3] − [3/4]} - In order to subtract the fraction, we need to have the same denominator, in this case 12
- x
^{[2/3] − [3/4]}= x^{[8/12] − [9/12]}= x^{ − [1/12]} - Eliminate negative exponents using the rule a
^{ − n}= [1/(a^{n})] - x
^{ − [1/12]}= [1/(x^{[1/12]})] - We need to eliminate the fractional exponent in the denominator.
- We need to multiply by a number (x
^{?}) such that when multiplied with (x^{[1/12]}) equals x - Multiply numerator and denominator by (x
^{[11/12]}) - [1/(x
^{[1/12]})]*( [(x^{[11/12]})/(x^{[11/12]})] ) = [(x^{[11/12]})/x]

^{[2/3]})/(x

^{[1/2]}− 3)]

- Recall that the rules of simplfying, which are:
- 1. No negative exponents
- 2. No fractional exponents in denominator
- 3. Not a complex fraction
- 4. Index as small as possible
- We are going to use these rules to guide us through the problem.
- Step 1: Multiply the numerator and denominator by the cojugate of the denominator. Use the shorcut a
^{2}− b^{2}. - [(x
^{[2/3]})/(x^{[1/2]}− 3)]*[(x^{[1/2]}+ 3)/(x^{[1/2]}+ 3)] = [(x^{[2/3]}(x^{[1/2]}+ 3))/((x^{[1/2]})^{2}− (3)^{2})] - Step 2: Simplify as much as possible
- [(x
^{[2/3]}(x^{[1/2]}+ 3))/((x^{[1/2]})^{2}− (3)^{2})] = [(x^{[2/6]}+ x^{[2/3]})/(x − 9)] = [(x^{[1/3]}+ x^{[2/3]})/(x − 9)]

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

### Rational Exponents

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

- Intro 0:00
- Definition 1 0:20
- Example: Using Numbers
- Example: Non-Negative
- Example: Odd
- Definition 2 4:32
- Restriction
- Example: Relate to Definition 1
- Example: m Not 1
- Simplifying Expressions 7:53
- Multiplication
- Division
- Multiply Exponents
- Raised Power
- Zero Power
- Negative Power
- Simplified Form 13:52
- Complex Fraction
- Negative Exponents
- Example: More Complicated
- Example 1: Write as Radical 19:03
- Example 2: Write with Rational Exponents 20:40
- Example 3: Complex Fraction 22:09
- Example 4: Complex Fraction 26:22

### Algebra 2

### Transcription: Rational Exponents

*Welcome to Educator.com.*0000

*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 2 ^{3}.*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 x ^{13}; 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 properties*0480

*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 a^{n} equals a^{m + n}.*0512

*And we did that working with things like x ^{3} times x^{2}.*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}/a^{n} 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^{1/6}z^{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 used*0709

*in earlier lessons, is that a ^{-n} = 1/a^{n}; 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 rational*0836

*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 fraction*0848

*So, if you have something like 2 divided by x, over y ^{4}, 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/a^{n}, 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/a^{n}, that means that x^{-3} is the same as 1/x^{3}.*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 x ^{1} in this case) equals x^{3/4}*0967

*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: x ^{1} = 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 x ^{1} = 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 check*1077

*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 something*1129

*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: (8c ^{2})^{3/5}.*1147

*Recall that a ^{m/n} = ^{n}√a^{m}.*1155

*Here, a equals 8c ^{2}; 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 8c ^{2} 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 c ^{6}; now this is in the simplest form that I can get it in, because this is equivalent to 8^{3} times (c^{2})^{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 = 3x ^{2} - 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--3x ^{2} - 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 3x ^{2} - 2 to the fourth, equals 3x^{2} - 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 x ^{1}, 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 multiply*1507

*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: a ^{2} - b^{2}.*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} - 2^{2}.*1720

*So, I am going to put that right down here: (x ^{1/2})^{2} - 2^{2}.*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 x^{1}.*1771

*So, it is just x; 2 squared is 4; this is going to give me 6 divided by 2, which is x ^{3}, 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 concept*1813

*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

2 answers

Last reply by: Kyoung-Hee Kim

Tue Oct 7, 2014 7:47 PM

Post by Kavita Agrawal on June 19, 2013

I don't think Example 1 is completely simplified. The 5th root of 512c^6 can be written as c times the 5th root of 512c, because c^6 = c^5 * c.

1 answer

Last reply by: Dr Carleen Eaton

Thu May 24, 2012 8:12 PM

Post by Darren Fuller on May 15, 2012

How would I solve a problem like this

2^5/2 - 2^3/2