Find the solution set of three to the 𝑥 power times three to the 𝑦 power equals 27 and three to the 𝑥 power plus three to the 𝑦 power equals 12.
Because of this “and” statement, we want values for 𝑥 and 𝑦 that make both of these sentences true. Starting on the left, I notice that 27 can be written as a power of three. 27 is three cubed. We bring the rest of the statement down. When we’re multiplying exponents with the same base, we know that we’re adding the exponents together. 𝑥 plus 𝑦 has to be equal to three. But there are a lot of things that make the statement 𝑥 plus 𝑦 equals three true.
To narrow down our 𝑥 and 𝑦, we’ll take a look at the equation on the right. We can’t write 12 as a base of three, but we do know that 12 equals four times three. Here, we also notice that we’re adding our two exponent values, not multiplying them. Four times three is like adding three to itself four times. Three added to itself three times equals nine. One way to break 12 down is to say 12 equals nine plus three.
Do you see where I’m going with this? Nine can be written with a base of three and an exponent. Nine can be written as three squared. Then we can write three as an exponent of three to the first power. This helps us narrow down what 𝑥 and 𝑦 could be.
One plus two equals three. 𝑥 could be one, and 𝑦 could be two. Two plus one is also equal to three. And that means 𝑥 could be two, and 𝑦 could be one. The solution said that would make both of these equations true is one, two and two, one.