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Determine the solution set of the square root of nine to the power of 𝑥 minus 18 times three to the power of 𝑥 plus 81 equals 72.
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This question is asking us to determine the solution set of an equation in 𝑥.
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In other words, we’re interested in finding the values of 𝑥 that make this equation true.
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Now this looks particularly nasty.
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But there is one simple thing that we can do to make it look a little bit nicer, that is, square both sides of the equation.
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By squaring the left-hand side of this equation, we’re essentially performing the inverse to square rooting.
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So we’re just left with the values inside the square root.
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That’s nine to the power of 𝑥 minus 18 times three to the power of 𝑥 plus 81.
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Of course, since we’ve squared the left-hand side, we need to repeat that on the right-hand side.
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And when we square 72, we get 5184.
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Next, we’ll subtract this value from both sides of our equation.
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Then we have an equation that’s equal to zero.
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That is, nine to the power of 𝑥 minus 18 times three to the power of 𝑥 minus 5103 is equal to zero.
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The key to solving this equation is spotting that nine can be written as three squared.
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And so we can write nine to the power of 𝑥 as three squared to the power of 𝑥.
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So our equation becomes three squared to the power of 𝑥 minus 18 times three to the power of 𝑥 minus 5103 equals zero.
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Now of course we can multiply these exponents.
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And we can say that three squared to the power of 𝑥 is the same as three to the power of two 𝑥.
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Then we can reverse this process and say, well, this must be the same as three to the power of 𝑥 squared.
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And so we write our equation as three to the power of 𝑥 squared minus 18 times three to the power of 𝑥 minus 5103 equals zero.
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And if we look carefully, we’ll see that this looks a little bit like a quadratic equation.
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And so we’re going to perform a substitution.
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We’re going to let 𝑦 be equal to three to the power of 𝑥.
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When we do, we find that we can write our equation as 𝑦 squared minus 18𝑦 minus 5103 equals zero.
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This equation is quite easy to solve.
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We need to factor the expression 𝑦 squared minus 18𝑦 minus 5103.
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We know that the first term in each binomial must be 𝑦 since 𝑦 times 𝑦 gives us 𝑦 squared as required.
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Then to find the other term in each binomial, we’re looking for two numbers whose product is negative 5103 and whose sum is negative 18.
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These numbers are negative 81 and 63.
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So our equation becomes 𝑦 minus 81 times 𝑦 plus 63 equals zero.
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And 𝑦 minus 81 and 𝑦 plus 63 are simply numbers.
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And their product is equal to zero.
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So for this to be the case, either 𝑦 minus 81 must be equal to zero or 𝑦 plus 63 must be equal to zero.
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Let’s solve this first equation for 𝑦 by adding 81 to both sides.
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So we find that 𝑦 is equal to 81.
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And we’ll solve this second equation by subtracting 63 from both sides.
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So 𝑦 is equal to negative 63.
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We’re not quite finished though.
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We were looking to solve this equation for 𝑥.
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So we go back to our earlier substitution 𝑦 is equal to three to the power of 𝑥.
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And we can replace 𝑦 in our solutions.
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And we see that either three to the power of 𝑥 is equal to 81 or it’s equal to negative 63.
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The value of 𝑥 that makes the equation three to the power of 𝑥 equals 81 true is four.
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But there are no values of 𝑥 such that three to the power of 𝑥 is equal to a negative value.
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So there are no solutions to our second equation.
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This means that the solution set of our equation consists of one number only.
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It’s four.