Solve the exponential equation. 8 1 x + 1 = 2 7 x + 5 81^{x+1}=27^{x+5} 8 1 x + 1 = 2 7 x + 5

Here, we're asked to solve the exponential equation. And here, we have 81 to the power of x plus 1 is equal to 27 to the power of x plus 5. Now this looks a little bit trickier in order to get the like bases on either side. Now 81, I know, is just 9 squared or 9 times 9, but 27 cannot be simplified into a power of 9 because it is 3 times 9. But looking at this, I know that 9 is just 3 squared, so 81 is really just 3 squared times 3 squared or 3 to the power of 4. And then 3 times 9, my 27, this just becomes 3 times 3 squared, which is really just 3 cubed. So I can actually rewrite these both as a base of 3. So this becomes 3 to the power of 4 to the power of x plus 1 is equal to 3 cubed to the power of x+5. Now I know that these have the same base, but let's ahead and rewrite these simplified a little bit, getting those exponents all put together. Now this 4 will multiply the x and the 1, and this 3 will multiply the x and the 5. So we need to be careful here. So this becomes 3 to the power of 4x+4 and that's equal to 3 to the power of 3x+15 distributing those exponents. Now that I have the same bases and I have simplified my exponents, I can go ahead and just set these exponents equal to each other, 4x+reexplus15. So rewriting that down here, I get 4x+4 is equal to 3x+15. And, again, I am left with a basic linear equation that I can solve. So I wanna pull all my x's to the same side, all of my constants to the other. So I'm going to go ahead and move this 3x over here and my 4 over here. So getting rid of that 3x, I subtract 3x from one side, canceling it, which means I need to subtract 3x from the other side. Now on this, I can subtract 4 canceling it, which whatever I do to one side, I have to do to the other. So here I'm left with 4 x minus 3 x, which is just x, and that's equal to 15 minuteus 4, which is 11. And we're completely done here. We're left with our answer. X is equal to 11. Now if I were to plug 11 back into my original exponential equation, I would end up with a true statement. Thanks for watching and I'll see you in the next one.