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# Simplifying hairy expression with fractional exponents

## Video transcript

let's get some practice simplifying hairy expressions that have exponents in them and so we have a hairy expression right over here and I encourage you to pause the video and see if you can rewrite this in a in a simpler way all right let's work through this together and the first thing that jumps out at me is the numerator here I have number 125 raised to the 1/8 power times the same number the same base 125 raised to the 5/8 power so I can rewrite this numerator I can rewrite this numerator using what I know of exponent properties as being equal to 125 to the sum of these two exponents to the negative 1/8 power plus 5/8 power all of that is going to be over the existing denominator we have which is five to the one-half all of that is going to be over five to the one half power so these are equivalent notice all I did is I added the exponents these two exponents cuz I had the same base and we were taking the product of both of these both 125 to the negative 1/8 and 125 to the 5/8 and so negative 1/8 plus 5/8 well that is 1/2 so this is this right over here is 1/2 so this is 125 to the 1/2 over 5 to the 1/2 well that's going to be the same thing this is going to be equivalent to 125 over 5 over 5 to the 1/2 power to the 1/2 power if I raise 125 to the 1/2 and I'm dividing by 5 to the 1/2 that's the same thing as doing the division first and then raising that to the 1/2 power well what's 125 divided by 5 well that's just 25 and what's 25 to the 1/2 well that's the same thing as the principal square root of 25 which is equal to 5 and we're all done that's simplified quite nicely let's do another one of these and this one is a little more interesting because we are starting to involve a variable we have the variable W but it's really going to be somewhat the same the same process and here the thing that jumps out at me is is the denominator I have the same base 3w squared 3w squared raised to one power one exponent times the same base 3w squared raised to another power so this is going to be equal to this is going to be equal to our numerator we can just rewrite it 12 w to the seventh power over negative three halves over our denominator we can write as this base 3w squared 3 w squared and we can add these two exponents so we could add negative two-thirds to negative 5/6 negative two-thirds to negative 5/6 well what is that going to be so let's see if I do negative negative 2/3 is the same thing as negative 4 6 minus 5/6 which is equal to negative 9 6 which is equal to negative 3 halves so this right over here is the same thing as negative 3 halves let me just write that negative 3 halves power negative 2/3 plus negative 5/6 is negative 3 halves now what's interesting is I have a negative three-halves up here and I have a negative three-halves over here so we can do the same thing we did in the last problem this could simplify to 12 w 12 w to the seventh power over 3 w squared 3 w squared all of that all of that to the all of that to the negative 3 half power notice what we did here had something to the negative 3 halves divided by something else to that negative three-halves well that's the same thing as doing the division first and then raising that that the quotient to the negative three-halves and what's nice about this this is pretty straightforward to simplify 12 divided by 3 is 4 and W to the 7th divided by W squared well we can divide both by W squared or you could say this is the same thing as W to the 7 minus 2 power so this is going to be W to the 5th power W to the fifth power and so it all simplified to four W to the fifth power to the negative three-halves to the negative three-halves now if we want to this is already pretty simple and at some point it becomes somewhat opinion someone's opinion on which expression is simpler than another and it might depend on what you're using the expression for but one could argue that you could keep trying to simplify this this is the same thing as this is the same thing as four to the negative three-halves to the negative three-halves x times W to the fifth to the negative three-halves and once again this is just straight out of our exponent properties now 4 to the negative three-halves let's just think about that four to the negative three-halves just in a different color just as a bit of an aside so 4 to the negative three-halves is equal to that's the same thing as 1 over 4 to the three-halves and that's C 4 to the the square root of 4 is 2 and we raise that to the third power it's going to be 8 so this is equal to 1/8 so that's equal to 1/8 and so all this is going to be equal to 1/8 and then W to the fifth and then that to the negative three-halves we can multiply these exponents that's going to be w to the 5 times negative three-halves what's going to be the negative 15 halves power so I don't know which one you would say is simpler this one over here or this one over here but they are equivalent and they both are a lot simpler than where we started