Video Transcript
Determine the solution set of π₯ to
the power of π₯ squared minus 64 equals six to the power of π₯ squared minus 64.
Now letβs begin by noticing that
the exponents on each side of our equation are in fact equal. And so, for both sides of our
equation to be equal, one value of π₯ would be when the bases, the big numbers, are
equal. In other words, if we have π₯ is
equal to six, both sides of our equation are identical, so that is one solution. But are there any other
options? Well, another way to ensure that
two sides of our equation are equal is to have a power of zero since anything to the
power of zero is one. And so what we could do is say that
the exponent π₯ squared minus 64 is equal to zero.
Letβs solve for π₯ by adding 64 to
both sides to give us π₯ squared equals 64. And then, finally, we take the
square root of both sides, remembering, of course, to take both the positive and
negative square root of 64. But since the square root of 64 is
eight, we can say that the solutions to this equation are π₯ equals negative or
positive eight. And so, so far, we have three
possible values of π₯. But in fact, there is one more. And that solution is π₯ equals
negative six. So, why does π₯ equals negative six
work?
Well, imagine π₯ is equal to
negative six. When it is, the exponent becomes
negative six squared minus 64, which is negative 28. And we know that if π is an even
number, the negative π₯ to the power of π is the same as π₯ to the power of π. And so, the left-hand side is
negative six to the power of negative 28. But because negative 28 is even,
this is the same as six to the power of negative 28, which is what we get on the
right-hand side. And so, π₯ equals negative six also
has to be a solution. Now, of course, we only chose
negative six because we wanted to match the bases. We wouldnβt go ahead and choose any
value of π₯ that would make the exponents equal. It only works for negative six.
And so we can say the solution set
to our equation is the set containing the elements six, negative six, eight, and
negative eight.