And to simplify this a little bit, we just have to remind ourselves that, if I raise something to one power, and then I raise that to another power, this is the same thing as raising my base to the product of these powers, a to the bc power. So, this is two to the 3x plus five power is equal to, instead of writing 64, I'm going to write two to the sixth power, two to the sixth power, and then that to the x minus seventh power, x minus seven power. This is the same thing as two to the sixth power, is 64, and I knew it was two to the sixth power because I just added the exponents because I had the same base. This was just a little bit easier for me.
Take six twos and multiply them together, you're going to get 64. Eight times eight is 64, so it's two to the sixth is equal to 64, and you can verify that. So, the key here is toĮxpress both of these with the same base, and lucky for us, 64 is a power of two, two to the, let's see, two to the third is eight, so it's going to be two to the You have two to the 3x plus five power, and then you have 64 to the x minus seven. Maybe 3x plus five needs to be equal to x minus seven,īut that wouldn't work, because these are two different bases. All right, so you might, at first, say, oh, wait a minute, Me what x is going to be, or what x needs to be to satisfy Once again, pause the video, and see if you can tell Let's say we have the exponential equation two to the 3x plus five power is equal to 64 to the x minus seventh power. Let's do another one of these, and let's make it a little bit more, a little bit more interesting. Divide both sides by nine, and we are left with x isĮqual to negative five. Say, well, 9x plus five needs to be equal to zero. Zero to the zeroth power we can discuss at some other time, but anything other than zero to the zeroth Anything to the zeroth power is going to be equal to one. Well, the key here is to realize that 26 to the zeroth power, to the zeroth power is equal to one. See if you can tell me what x is going to be. We have 26 to the 9x plus five power equals one. Let's get some practice solving some exponential equations, and we have one right over here.
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