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# Worked example: Exponent properties

## Video transcript

simplify 25a to the third and that a to the third is being raised to the third power times B squared all of that over five 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 that 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 to the 3 times 3 power or a to the 9th 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 1 plus 2 power which is equal to B to the third power and then the last thing that 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 is our expression simplified to we have 5a to the 9th a to the 9th and then we still have this B squared here B squared all of that over a squared 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 9th I'm just in a slightly different color we have an a to the 9th over a squared what's that going to simplify to well that's going to be the same thing let me write this a to the ninth over a squared is the same thing as a to the 9 minus 2 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 to 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 like that right now so this whole expression simplifies to it simplifies to five times a to the seventh power because this simplifies to a to the seventh a to the seventh times and all the B's right here simplified to B to the negative one B to the negative one we could leave it like that and you know that's pretty simple but we might not want a negative exponent there and then we just have to remember that B to the negative one power is the same thing as 1 over B and if we remember that then we can rewrite this entire expression as in the numerator we'll have a 5 and we'll have a to the seventh 5 a to the seventh and then in the denominator we'll have the B so we're multiplying this times 1 over B so that's the same thing as B to the negative 1 and we are done