Video: Evaluating Algebraic Expressions after Solving Exponential Equations

Given that 8^𝑦 = 4^𝑧 = 64, find the value of 𝑦 + 𝑧.

03:26

Video Transcript

Given that eight to the power of 𝑦 equals four to the power of 𝑧 equals 64, find the value of 𝑦 plus 𝑧.

The first stage of solving this problem is gonna be finding 𝑦. So what I’ve done is I’ve actually put eight to the power of 𝑦 is equal to 64. So therefore I can say that 𝑦 is equal to two. And we know that because we know that eight squared is equal to 64 or eight is actually the root of 64.

Okay, great! So now we’ve found 𝑦. Let’s move on and find 𝑧. There are actually a couple of ways we could find 𝑧. What I’m gonna do is I’m gonna use one method now and then use another method to actually check our answer. So first of all, we have eight to the power of 𝑦 is equal to four to the power of 𝑧.

And what I’m gonna do is I’m gonna actually use these two relationships to actually make them have the same base number, because what we have is eight is equal to two to the power of three and four is equal to two to the power of two or two squared. So therefore, what we actually have is two to the power of three to the power of 𝑦 is equal to two squared or two to the power of two to the power of 𝑧.

So what I’m gonna do now is I’m actually gonna use one of our exponent rules to actually simplify even further, because what we have is π‘₯ to the power of π‘Ž to the power of 𝑏 is equal to π‘₯ to the power of π‘Žπ‘. So we actually multiply the exponents. So therefore, we have two to the power of three 𝑦 is equal to two to the power of two 𝑧. And that’s because we multiply three and 𝑦 and two and 𝑧.

And now because we’ve actually got the same base number either side of our equation, we can now equate the exponents. So therefore, we have three 𝑦 is equal to two 𝑧. So at this point, what we can do is actually substitute back in our value for 𝑦. And we know that 𝑦 is equal to two. So we’re gonna get three multiplied by two is equal to two 𝑧.

So therefore, we have a value of 𝑧 which is equal to three. And we get that because we have three multiplied by two, so we divide each side by two. Then we’re left with three. And as I said, we can check that using the other method that we could’ve used to find 𝑧.

And we have four to the power of 𝑧 is equal to 64. So therefore, 𝑧 will be equal to three as we know that four to the power of three is equal to 64 because four multiplied by four is 16 multiplied by another four is 64.

Okay, great! So we found 𝑦 and 𝑧. And we’ve checked 𝑧. So now we just move on to the final part of the question, which to find 𝑦 plus 𝑧. So we’ve got 𝑦 plus 𝑧. And then we substitute our value of 𝑦 equals two and our value that 𝑧 equals three.

So we get two plus three. So therefore, we can say that given that eight to the power of 𝑦 equals four to the power of 𝑧 equals 64, then the value of 𝑦 plus 𝑧 is equal to five.

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