# Question Video: Finding the Solution Set of an Exponential Equation Mathematics

Determine the solution set of 6^(π₯) β 3^(π₯ β 1) Γ 2^(π₯ β 6) = 573/16.

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### Video Transcript

Determine the solution set of six to the power of π₯ minus three to the power of π₯ minus one times two to the power of π₯ minus six equals 573 over 16.

By determining the solution set, weβre looking for all the values of π₯ that satisfy our equation. Now, before we can do anything with this equation, itβs really sensible to perform some manipulation. And the first thing weβre going to recall is one of our laws of exponents. And this says that π₯ to the power of π divided by π₯ to the power of π is π₯ to the power of π minus π. In other words, when dividing two numbers whose base is the same, we subtract their exponents. And this means three to the power of π₯ minus one is the same as three to the power of π₯ divided by three to the power of one or three to the power of π₯ divided by three.

Similarly, two to the power of π₯ minus six is the same as two to the power of π₯ divided by two to the power of six. And so we rewrite our equation as six to the power of π₯ minus three to the power of π₯ over three times two to the power of π₯ over two to the power of six equals 573 over 16. Now, in order to make this entire equation look a little bit nicer, weβre going to multiply through by three times two to the power of six, which actually is 192. And when we do, we get 192 times six to the power of π₯ minus three to the power of π₯ times two to the power of π₯ equals 6876.

Now actually, three to the power of π₯ times two to the power of π₯ can be written as three times two all to the power of π₯, which is, of course, six to the power of π₯. And so our equation is 192 times six to the power of π₯ minus six to the power of π₯ equals 6876. Really, we want to make π₯ the subject or at least for now six to the power of π₯. So we factor the expression on the left-hand side such that six to the power of π₯ times 192 minus one is 6876, or 191 times six to the power of π₯ equals 6876.

Next, letβs divide through by 191. That gives us six to the power of π₯ equals 36. Well, we know that six squared is 36, so that means π₯ is equal to two. And so there is one solution to our equation, and thatβs two. Our solution set is two.