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CCSS.Math: , ,

let's get even more practice solving some exponential equations and I have two different exponential equations here and like always pause the video and see if you can solve for X in both of them alright let's tackle this one in purple first and you might first notice that on both sides of the equation I have different bases so it would be nice to have a common base and when you look at it you're like well 32 is not a power of 8 or at least it's not an integer power of 8 but they are both powers of 2 32 is the same thing as 2 to the fifth power 2 to the fifth power and 8 is the same thing as 2 to the third power 2 to the third power so I can rewrite our original equation as instead of writing 32 I could write it as 2 to the fifth and then that's going to be raised to the x over 3 power x over 3 power is equal to instead of writing 8 I could write 2 to the third power 2 to the third power and I'm raising that to the X minus 12x minus 12 now if I raise something to a power and then raise that to a power I could just multiply these exponents so I could rewrite the left-hand side as 2 to the 5 x over 3 5x over 3 power I just multiply these exponents and that's going to be equal to 2 to the and now I just multiply the 3 times X minus 12 so 2 to the 3x minus 36 and now things have simplified nicely I have 2 to this power is equal to 2 to that power so these two exponents must be equal to each other 5x over 3 must be equal to 3x minus 36 so let's set them equal to each other and solve for x so 5x over 3 is equal to 3x minus 36 let's see we could we could multiply we could multiply everything I three let's do that so if we multiply everything times three you're going to get 5x is equal to 9x minus this 9x minus 108 and now we can subtract 9x from both sides and so we will get 5x minus 9x is going to be negative 4x is equal to negative 108 we're in the homestretch here divide whoops sorry about that we could divide both sides by negative 4 negative 4 and we are left with X is equal to what is this going to be 27 X is equal to 27 and we are all done we're all done and if you substitute it X back in there you would get 32 to the 27 divided by 3 so 30 to the 9th power is the same thing as 8 to the 27 minus 12th power so 8 to the fifteenth 8 27 minus 12 8 to the 15th power so anyway that was that was fun let's do that let's do the next one now so this one looks interesting in other ways we have rational expressions we have an exponential appear exponential down here and the key realization here is well the first thing I'd like to do let me put this let me write this 25 in terms of 5 we know that 25 is the same thing as 5 squared so we can rewrite this as 5 to the 4x plus 3 over instead of 25 I could rewrite that as 5 squared and then I'm going to raise that to the 9 minus X to the 9 minus X and that of course is going to be equal to 5 to the 2x + 5 now 5 to the second and then that to the 9 X 9 minus X I can just multiply these exponents so this is going to be 5 to the 4x plus 3 over 5 to the 2 times 9 is 18 2 times negative x is negative if 2x and that is going to be equal to that is going to be equal to 5 to the 2x plus 5 and now let's see there's multiple ways that we could tackle it we could multiply both sides of this equation by 5 to the 5 to the 18 minus 2x that's one way to do it or we could say hey look I have 5 to some exponent divided by 5 to some other exponent so I could just subtract this blue exponent from this yellow one so the left hand side will simplify to 5 to the 4x + 3 - we just - in a neutral color - 18 minus 2x 18 minus 2x and that of course is going to be equal to what we've had on the right hand side 5 to the 2x + 5 now we just have to simplify a little bit let's see this is going to be in fact we could just say look I'm having trouble with my little pen tool whoops all right so now we could say this exponent needs to be equal to that exponent because we have the same base and so what we have here on the left hand side that I can rewrite as 4x plus 3 minus 18 plus 2x I'm just multiplying the negative times both of these terms so plus 2x is going to be equal to 2x plus 5 2x plus 5 so there's a bunch of different things we could do here 1 we could subtract 2x from both sides that'll clean it up a little bit 2x from both sides we could also subtract 5 from both sides so let's just do that so well well I let me just subtract it subtract 5 from both sides I'm skipping some steps here but I figure you're you're at this point reasonably comfortable with linear equations so then on the left hand side we are going to have 4x and then you have 3 minus 18 minus 5/3 minus 18 is negative 15 minus 5 is negative 20 is going to be equal to 0 and then because let's cancel out and so add 20 to both sides you get 4x is equal to 20 divide both sides by 4 and we get X is equal to 5 and we are all done