Video Transcript
Given that eight to the power of 𝑥 equals four to the power of 𝑦, find the value of 512 to the power of 𝑥 over 𝑦 plus 64 to the power of 𝑦 over 𝑥.
In order to answer this question, it might be helpful to spot that every single number eight, four, 512, and 64 can be written as a power of two. In particular, eight can be written as two cubed and four can be written as two squared. And this is really useful because we can create an equation in which the bases are the same. And then we can simply compare the exponents. To find an expression for eight to the power of 𝑥 with the base of two, we rewrite it as two cubed to the power of 𝑥. And then we recall that since we’re working with an expression in parentheses, we simply multiply the exponents. So eight to the power of 𝑥 is two to the power of three 𝑥.
Similarly, we’re going to write four to the power of 𝑦 as two squared to the power of 𝑦. Then multiplying the exponents, and we find that four to the power of 𝑦 is the same as two to the power of two 𝑦. Let’s replace eight to the power of 𝑥 and four to the power of 𝑦 with our new expressions. When we do, we find that two to the power of three 𝑥 equals two to the power of two 𝑦. Now, since the base is the same for these two expressions to be equal, their exponents must also be the same. So three 𝑥 equals two 𝑦. And this is great because we can now find a value for 𝑥 over 𝑦 and 𝑦 over 𝑥.
To find the value of 𝑥 over 𝑦, we’ll divide both sides of the equation by 𝑦 and by three. So 𝑥 over 𝑦 is two-thirds. In a similar way, to find the value of 𝑦 over 𝑥, we’ll divide both sides by 𝑥 and by two. So 𝑦 over 𝑥 is equal to three over two. So we now replace 𝑥 over 𝑦 and 𝑦 over 𝑥 with these fractions. So we’re trying to find the value of 512 to the power of two over three plus 64 to the power of three over two.
Now, we might recall that for positive bases, 𝑥 to the power of 𝑎 over 𝑏 is found by taking the 𝑏th root of 𝑥 and raising that to the power of 𝑎. So in the case of our expression, we get the cube root of 512 squared plus the square root of 64 cubed. But in fact, the cube root of 512 is eight, as is the square root of 64. So this is eight squared plus eight cubed. That’s 64 plus 512 which is equal to 576. So given that eight to the power of 𝑥 equals four to the power of 𝑦, the value of 512 to the power of 𝑥 over 𝑦 plus 64 to the power of 𝑦 over 𝑥 is 576.
