Most people overthink this exponential math problem.

To solve this, we need to first of all put everything in base 2. So this becomes 2 to the power of n plus 4 is 2 to the power of 2. So we have 2 to the power of 2 raised to the power of n plus 8 is 2 to the power of 3. 2 to the power of 3 raised to the power of n is equal to 39. Welcome back to my channel. So this is where we are right now. Now recall that a to the power of n raised to the power of n can also be written as a to the power of n raised to the power of n. That means that we can write all this as 2 to the power of n plus 2 to the power of n raised to the power of 2 plus 2 to the power of n raised to the power of 3 is equal to 39. Now since we have 2 to the power of n here, 2 to the power of n here, and 2 to the power of n here as well, we can say let 2 to the power of n be equal to x. Therefore this equation becomes x to the power of n plus oh sorry this equation becomes x plus x squared plus x cubed is equal to 39. Please subscribe to my channel and turn on your notification bell if this is the first time you are seeing this wonderful channel. We upload our videos as often as possible. Tune in as you are convenient. There must be something new for you to watch. Give us a thumbs up if you love what we are doing and please comment. Very important. Tell me what you feel about my video. Things I need to adjust. Where I'm doing well and where I'm not doing well. I will be appreciating you. Okay now back to this. We can look for a number that when we add the number to each square and we get 39. Okay and if we check that number is 3 because 3 plus 9 plus 27 is 39. Therefore we can write our experiment x plus x squared plus x cubed is equal to now this 39 becomes 3 plus 9 plus 27. So moving forward we have x plus x squared plus x cubed is equal to 3 plus 9 is 3 squared plus 27 is 3 cubed. Now we are going to bring this to the other side in pairs. Okay according to their powers. So I'll write x and I'll bring 3 as minus 3 plus x squared and I'll bring 3 squared as minus 3 squared plus x cubed and I'll bring this as minus 3 cubed is equal to 0. Now recall that a squared minus b squared is equal to a minus b multiplied by a plus b and a to the power of 3 minus b to the power of 3 is equal to a minus b into a squared plus a b plus b squared. Yes therefore we are going to expand this and this with respect to this. So we go ahead and write that. I need to erase this. Yes I need to. Have you subscribed yet? If you have not please go ahead. If not this might be the last time you'll be seeing this wonderful channel. Okay so we continue. Now we have x minus 3 into okay plus brackets x squared minus 3 squared in this form will be x minus 3 multiplied by x plus 3 plus and this in this form will be x minus 3 multiplied by x squared plus 3 x plus 3 squared. Now everything is equal to 0. Okay if you look closely you're going to notice x minus 3 x minus 3 x minus 3 meaning that we can actually factorize that right. So let's do that x minus 3 into when this is divided by itself we have 1 plus when this is divided by this this will remain plus when this is divided by this this will remain. 3 squared is 9 is equal to 0. Okay so now we have x minus 3 brackets 1 okay let's start with x squared then plus x plus 3 x is plus 4 x plus 1 plus 3 plus 9 is 13 plus 13 right is equal to 0. Now from here we can see that x minus 3 equals 0 or x squared plus 4 x plus 13 is equal to 0. Moving forward okay so I'm going to write this here that x minus 3 is 0 or x squared plus 4 x plus 13 is also 0. Alright this way gets interesting and I want you to pay attention. Okay now we have x is equal to 3 from here or from here using the quadratic equation formula we can say that x from here our a is 1 our b is plus 4 and our c is 13. So using the formula that x is minus b plus or minus square root of b squared minus 4ac all over 2a. We can say that x will be minus 4 plus or minus square root of 4 squared minus 4ac 4 multiplied by 1 multiplied by 13 everything divided by 2 multiplied by 1. Now you will see from here that clearly this is actually I prefer using like damp clothes to erase this because this gives me black marks on my board. So I have x is equal to negative 4 plus or minus square root of 4 squared is 16 minus 4 times 13 is um that should be 72 right 4 times 3 12 carry 1 4 times 1 is 4 plus 1 is 5 or 52. Alright now you can see that from here I don't need to go further because this take away this will give me a negative answer meaning that the values of x here are going to be complex and we don't need that all right therefore we are going to continue with x equals 3. Remember we are looking for n and not x and definitely we are going to prove that the answer we got is correct don't miss that part that part is very interesting trust me. So moving forward now I can say x equals 3 but recall where we said let 2 to the power of n be equal to x right so meaning that we'll write 2 to the power of n we are looking for n is equal to 3. Now is this actually possible? Yeah it's possible if we can introduce logarithm to both sides of this equation so let's go ahead and do that so introduce log 2 to the power of n is equal to log 3. Now remember that log a raised to the power of b is equal to b multiplied by log a therefore we can write this as n log 2 is equal to log 3. Now we need to divide both sides of this equation by log 2 so divided by log 2 divided by log 2. Now log 2 we go into this remember we are going to prove that the answer we got is correct so from here we have n is equal to now log a over b with change of base right is log 3 base 2 right so we have log 3 base 2. Now let us prove that the answer we got is correct so proof we have that 2 to the power of n which is log 3 base 2 plus 4 is 2 right 2 to the power of 2 brackets n is log 3 base 2 plus 8 is 2 to the is log 3 base 2 so log 3 base 2. Now remember that log a remember that a raised to the power of log b base a is also equal to b all right a raised to the power of log b base a is b therefore we can actually say that this and this are the same so the whole of this is 3 plus 2 okay for this one this 2 can be written as log 3 to the power of 2 this 2 can come up right so this can be written as 2 to the power of log 3 squared base 2 plus and this is 2 to the power of log 3 to the power of 3 right base 2. Now is this actually equal to 39 we are proving that so using my the law that I explained earlier we can now write this as 3 plus this and this the same so the whole of this is 3 squared which is 9 plus then this and this the same the whole of this is 3 to the power of 3 which is 27. Now is this equal to 39 yes we have 37 plus 3 is 30 plus 9 is 39 so we can see that the answer we got is correct. Thank you so much for watching and see you in my next video. Bye.