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CCSS.Math: , ,

let's get some practice solving some exponential equations and we have one right over here we have 26 to the 9x plus 5 power equals 1 so pause the video and see if you can tell me what X is going to be well the key here is to realize the 26 to the 0th power to the zeroth power is equal to 1 anything to the zeroth power is going to be equal to 1 0 to 0 power we can discuss it some other time but anything other than 0 to the zeroth power is going to be 1 so we just have to say well 9x plus 5 needs to be equal to 0 9x plus 5 needs to be equal to 0 and this is pretty straightforward to solve subtract 5 from both sides and we get 9x is equal to negative 5 divide both sides by 9 and we are left with X is equal to negative 5 let's do another one of these and let's let's make it a little bit more a little bit more interesting let's say we have the exponential equation 2 to the 3x plus 5 power is equal to 64 to the X minus 7th power once again pause the video and see if you can tell me what X is going to be or what X needs to be to satisfy this exponential equation alright so you might at first say oh it may be 3x plus 5 needs to be equal to X minus 7 but that wouldn't work because these are two different bases you have 2 to the 3x plus 5 power then you have 64 to the X minus 7 so the key here is to express both of these with the same base and lucky for us 64 is a power of 2 2 to the let's see 2 to the third is 8 so it's going to be 2 to the third times 2 to the third 8 times 8 is 64 so it's 2 to the sixth is equal to 64 and you can verify that take six 2s and multiply them together you're going to get 64 this is just a little bit easier for me eight times eight and this is the same thing as two to the sixth power is 64 and I knew it was to the six power because I just added the exponents because I had the same base all right so I can rewrite 64 all right let me rewrite the whole thing so this is 2 to the 3x plus 5 power is equal to instead of writing a 64 I'm going to write 2 to the sixth power 2 to the sixth power and then that to the X minus seventh power X minus 7 power 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 equation I can rewrite as 2 to the 3x plus 5 is equal to 2 to the and I just multiply 6 times X minus 7 so it's going to be 6x 6x minus 6 times 7 is 42 I'll just write the whole thing in yellow so 6x minus 42 I just multiply the 6 times the entire expression X minus 7 and so now it's interesting I have 2 to the 3x plus 5 power has to be equal to 2 to the 6 X minus 42 power so these need to be the same exponent so 3x plus 5 needs to be equal to 6x minus 42 so there we go it sets up a nice little linear equation for us 3x plus 5 is equal to 6x minus 42 let's see we could get all of our since I'll put my all my X's on the right-hand side since I have more X's on the right already so let me subtract 3x from both sides and let me I want to get rid of this 42 here so let's add 42 to both sides and we are going to be left with five plus 42 is forty seven is equal to forty seven is equal to 3x now we just divide both sides by three and we are left with X is equal to 47 over three X is equal to 47 over three and we are done