Using laws of exponents
Worked example: Exponent properties
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!