Question Video: Solving Exponential Equations Using Laws of Exponents

Video Transcript

Given that three to the power of 𝑥 equals 27 and three to the power of 𝑥 plus 𝑦 equals 81, find the value of 𝑥 and the value of 𝑦.

Well, there’re a couple of ways we go about solving it. One way is to use an exponent rule. And that rule is a multiplication rule, which tells us that if we have 𝑥 to the power of 𝑎 multiplied by 𝑥 to the power of 𝑏, it’s equal to 𝑥 to the power of 𝑎 plus 𝑏. So therefore, using this, what we can do is rewrite three to the power of 𝑥 plus 𝑦 equals 81 as three to the power of 𝑥 multiplied by three to the power 𝑦 is equal to 81.

And now, what we can do is we can substitute in three to the power of 𝑥 equals 27 into our equation. And if we do that, what we get is 27 multiplied by three to the power of 𝑦 equals 81. So, now what we can do is divide both sides of our equation by 27 to find out what three to the power of 𝑦 is. When we do that, we get three to the power of 𝑦. And that’s because if we divide the left-hand side by 27, we’re left with three to the power of 𝑦. And we get it’s equal to three, and that’s because 81 divided by 27 is three.

Well, if three to the power of 𝑦 is equal to three, therefore 𝑦 must be equal to one. And that’s because three to the power of one is three. Also, we can think about it as if we equate our exponents at this point, we’d have 𝑦 is equal to one. So, that’s our value of 𝑦 found. And now, we want to find our value of 𝑥. Well, we can say that three multiplied by three multiplied by three is equal to 27. So therefore, three cubed is equal to 27. So therefore, we can say that 𝑥 is equal to three and 𝑦 is equal to one.

What we can do is quickly check this. And we can do that by substituting in 𝑥 equals three and 𝑦 equals one into three to the power of 𝑥 plus 𝑦 equals 81. So, we do that, we’re gonna get three to the power of three plus one equals 81, which gives us three to the power of four is equal to 81. Or, three multiplied by three is nine multiplied by another three is 27 then multiplied by a fourth three is 81. So yes, this is correct. We can definitely say that 𝑥 equals three and 𝑦 equals one.