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### Course: Math (NSDC) - English > Unit 7

Lesson 1: Properties of exponents# Worked example: Exponent properties

Use exponent properties to simplify a challenging expression. Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

- Would the answer still be correct if it is written as 5a^7*b^-1 rather than the fraction form?(5 votes)
- Yes that would be correct too.

But completely simplified would be in fraction form.(5 votes)

- with the end result, 5a to the power of 7 over b, could you still put it as 5b over a to the power of 7? if not, why does a to the power of 7 have to be over b?(2 votes)
- The positive exponents go in the numerator and the negative exponents go in the denominator. That is why 5 is on top with a^7 and b is on the bottom. If you wanted to move b to the numerator along with 5a^7, you would have to write it as b^-1, to indicate it is a negative exponent, as Sal did. Otherwise, if you write it as b and leave it in the numerator , people will assume it's a positive exponent,as positive exponents go in the numerator. 5b/a^7 would mean b is a positive exponent (which it is not) and a^7 is a negative exponent, which it is not. Putting 5 in the numerator, however, would be acceptable, but putting b in the numerator is not (unless written with a negative exponent of 1).(4 votes)

- When I type this 6/5y^2 the answer shows 6/5 with the y^2 not being in the denominator and instead being on the right of the whole 6/5. How do I type it to where it goes in the denominator?(3 votes)
- Type:
`6 / (5y^2)`

It is very important to use parentheses when typing these expressions, to make sure the software knows what you mean. There is no harm in adding extra parentheses just to be sure.(2 votes)

- Like what Sal did in1:45with the a^9/a^2 turned into a 9^7. So if I had a b^4/b^2 then i would come out with a b^2. right?(3 votes)
- When the is 5a to the 7th power over b, and b was to the negative 1st power, does that mean that the number with the negative exponent goes on the bottom of the fraction when simplified?(2 votes)
- Is this the problem you mean?

5a^7 / b^-1

The b^-1 in the denominator can be changed to b^1 in the numerator.

THen you have:

5a^7b^1

If you have a negative power in the numerator, it becomes positive in the denominator. If you have a negative power in the denominator, it becomes positive in the numerator.(2 votes)

- At3:03I notice that the final product of the equation is 5a^7/b but could we put a 1 after 5a^7 so that it looks like 5a^7 1/b? I am a bit confused so please forgive me if what I write doesn't add up to what is actually in the seminar. and if this is possible, can I then add a + symbol so that it looks like 5a^7+1 over b??

Thanks in advance for the help!(2 votes)- you are correct that 5a^7/b is the same as (5a^7)1/b. But No, you can not use a plus sign. It would be 5 times (a^7) times (1/b). or 5*(a^7)*(1/b).

I hope that helps.(2 votes)

- So if you can flip a part of a denominator with a negative exponent, what do you flip? just exponent? Base? Coefficents?(2 votes)
- 24^3= 13824

Because if you do 24 times 24 times 24 you get 13824(1 vote)

- When the is 5a to the 7th power over b, and b was to the negative 1st power, does that mean that the number with the negative exponent goes on the bottom of the fraction when simplified?(2 votes)
- Yes, go back to the definition of negative exponents. a^-1 = 1/a and 1/a^-1 = a. Regardless of whether the negative exponent is on the top of bottom, you still take the reciprocal and make the exponent positive.(2 votes)

- @3:07Why did the denominator get the B?(2 votes)
- Because there is a b to the first in there B^1*B^2=B^3 however there is only a B^2 on top which means that you B^-1 which ends up in the denominater(2 votes)

- Do you know if there is a video about adding and subtracting exponents?(2 votes)
- I can summarize some of the rules of exponents and tell you that.......

1.)we add two exponents when we multiply two or more bases raised respectively by to a certain power n. Mathematically:

a^n * a^m = a^(n+m)

for instance using real numbers......

3^2 * 3^5 = 3^(2+5) = 3^7

2.) we subtract two exponents when we divide two or more bases raised respectively by to a certain power n or when we multiply bases at a negative power. Mathematically:

a^n / a^m = a^n * a^-m = a^(n-m)

using real numbers.......

3^2 / 3^5 = 3^2 * 3^-5 = 3^(2-5) = 3^-3

You also can watch this and the other videos next to it at:

https://www.khanacademy.org/math/algebra/exponent-equations/exponent-properties-algebra/v/exponent-properties-1

https://www.khanacademy.org/math/algebra/exponent-equations/exponent-properties-algebra/v/exponent-properties-4?qa_expand_key=ag5zfmtoYW4tYWNhZGVteXJpCxIIVXNlckRhdGEiTHVzZXJfaWRfa2V5X2h0dHA6Ly9ub3VzZXJpZC5raGFuYWNhZGVteS5vcmcvOTU0ZjEzMzc2MmZhMTU4ZThiNGUzNWY0YTNhMWNiNDAMCxIIRmVlZGJhY2sYiScM(2 votes)

## Video transcript

Simplify 25 a to the third and a to the third is being raised to the third power, times b squared and all of that over 5 a squared, b times b squared So we can do this in multiple ways, simplify different parts. What I want to do is simplify this part right over here. a to the third power, and we're raising that to the third power. So this is going to be from the power property of exponents, or the power rule this is going to be the same thing as a to the 3 times 3 power So this over here (let me scroll up a little bit) is going to be equal to a the 3 times 3 power, or a to the ninth power. We could also simplify this b times b squared over here. This b times b squared, that is the same thing as b to the first power remember, b is just b to the first power. So it's b to the first power times b to the second power. So b to the first times b to the second power is just equal to b to the one plus two power, which is equal to b to the third power and then last thing we could simplify, just right off the bat just looking at this: we have a 25 divided by 5. Well that's just going to give us 5. or we could say it's going to give us 5 over 1 if you view it as dividing the numerator and the denominator both by 5. So what does our expression simiply to? We have 5a to the ninth, and then we still have this b squared here, b squared. All of that over a squared times b to the third power... times b to the third power. Now, we can use the quotient property of exponents. You have an a to the ninth Let me use a slightly different color. We have an a to the ninth. over a squared. What's that going to simplify to? Well, that's going to simplify to be the same thing, let me write this a to the ninth over a squared, the same thing as a to the nine minus two, which is equal to a to the seventh power. Now we also have and this will get a little bit interesting here We have a b squared over b to the third power So that simplifies too. So b squared over b to the third is equal to b to the two minus three power, which is equal to b to the negative one power. And we'll leave it alone like that right now. So this whole expression simplifies to It simplifies to: 5 times a to the seventh power (because this simplifies to a to the seventh) a to the seventh, times (the bs right here simplify too) b to the negative one. We could leave it like that, you know, that's pretty simple but we may not want a negative exponent there we just have to remeber that b to the negative one power is the same thing as one over b. Now if we remember that, then we can rewrite this entire expression as, the numerator will have a five and will have a to the seventh 5 a to the seventh. And then the denominator will have the b. So we're multiplying this times one over b. That's the same thing as b to the negative one. And we are done!