Given that two to the 𝑥 power equals eight to the 𝑦 power which equals 512, determine the value of 𝑥 minus 𝑦.
If we know that two to the 𝑥 power equals eight to the 𝑦 power which equals 512, we can break this into two smaller statements. Two to the 𝑥 power equals 512. Eight to the 𝑦 power equals 512.
In order to solve a problem like this, we want to rewrite 512 with the base of two and a base of eight. We need to ask the question, eight to the what power would equal 512? In order to do that, we should divide 512 by eight. Eight goes into 51 six times. Six times eight is 48, which means there’s a remainder of three. And eight goes into 32 four times with no remainder because four times eight is 32. This tells us that eight times 64 equals 512. And we recognize 64 as eight squared, eight times eight. And so we can say that eight times eight times eight equals 512, that eight cubed equals 512.
And if eight cubed is equal to 512, then eight to the 𝑦 power is equal to eight cubed. Therefore, 𝑦 equals three. We could repeat this process to solve for two. We could divide 512 by two. However, since we started with the eights, we can use this to more quickly solve for two to the what power equals 512. We can do this because we know eight equals two times four. And four equals two times two. We could say that eight is equal to two cubed. And therefore two cubed times two cubed times two cubed equals 512.
And to simplify this, we know that 𝑎 to the 𝑥 power times 𝑎 to the 𝑦 power equals 𝑎 to the 𝑥 plus 𝑦 power. And that means, to multiply these three values together, we will add their exponent values. Three plus three plus three equals nine, and 512 equals two to the ninth power. So we substitute two to the ninth power in for 512. And we get two to the 𝑥 power equals two to the ninth power. Therefore, 𝑥 equals nine. Remember, our whole goal here is to find 𝑥 minus 𝑦. 𝑥 minus 𝑦 equals nine minus three. And so 𝑥 minus 𝑦 equals six.