Solving an exponential equation with negative base!

Added: 2026-05-07

[00:00:00]Let's see how we are going to solve this equation. We have 2 to the x is equal to parentheses with -64 inside raised to the square root of x power. And this is unusual because we have a negative base here.

[00:00:13]Mhm.

[00:00:14]But I will tell you though, for this one in particular, we still have good answer. It's just real numbers by the way.

[00:00:20]So, if you haven't done so already, please pause the video and try this first before you watch the solution.

[00:00:25]Done? Here we go.

[00:00:27]What I'm going to do is I will actually look at this right here as -1 * 64 and then raised to square root of x power.

[00:00:38]And then because the inside is just multiplying, so I can put this to square root of x power and then times 64 to square root of x power. And then this is quite nice because 64 happens to be 2 to some power. What power? The sixth power. So, let's write that and then raised to the square root of x power.

[00:01:05]And then we can just multiply the exponents here.

[00:01:09]And we get 2 to the sixth square root of x power.

[00:01:15]So, that's just us cleaning up for now, but how can we continue though?

[00:01:20]Here's the key.

[00:01:21]-1 to the square root of x power. Right here, we're just going for real solutions.

[00:01:27]So, when we have -1 to some power, we really want to just have this part being equal to 1.

[00:01:36]And the reason is because when we have 2 to the x, the output is always positive.

[00:01:41]Likewise, 2 to this power is always going to be positive.

[00:01:45]If this gives us a negative, then we get positive is equal to a negative number, which is not possible. So, what we want to do is let's just go ahead and solve for x from here being equal to that.

[00:01:59]And then later on we'll check the answer.

[00:02:04]This part is nice because both sides have base 2. So, that means the exponents have to be equal, so we get x equals 6 square root of x.

[00:02:16]Squaring both sides, we get x squared equals 6 squared is 36. Square root of x squared is just x.

[00:02:25]Right here, be sure to not divide both sides by x because otherwise, you'll be missing zero as the one of the solutions.

[00:02:33]So, bring this to the other side instead.

[00:02:38]And notice we have the x, so we can factor the out.

[00:02:44]From here, we see that x is equal to zero. From here, we see that x - 36 has to be zero, so that means x has to be 36.

[00:02:54]Once we work this out, we have two possible answers that we can check.

[00:02:58]Remember, we want it to be one for this part. So, if we plug in zero into the square root here, we get -1 raised to the square root of zero's power.

[00:03:10]That's just -1 to the zero's power and we do end up with one. Checks.

[00:03:16]And if we plug in 36 into here, -1 to the square root of 36 power is just -1 to the sixth power and we also end up with one. So, that also checks. So, in fact, both of these are solutions to that equation.

[00:03:35]Now, imagine if we have to solve 2 to the x is equal to -8 raised to the square root of x power.

[00:03:46]Well, I will leave this one for you guys. Go ahead and leave a comment down below and let me know what the answer is.