Given that 𝑥 is equal to negative one-third, 𝑦 is equal to negative one-half, and 𝑧 is equal to three-halves, find the numerical value of six 𝑥 squared 𝑦 squared 𝑧 cubed.
So in order to find the numerical value of six 𝑥 squared 𝑦 squared 𝑧 cubed, we must first substitute the values of 𝑥, 𝑦, and 𝑧 in. So we will substitute negative one-third in for 𝑥, negative one-half in for 𝑦, and three-halves in for 𝑧. And when we write this, six and 𝑥 squared and 𝑦 squared and 𝑧 cubed are all being multiplied together. So we can write six times 𝑥 squared times 𝑦 squared times 𝑧 cubed.
And remember that when a fraction is raised to a power, we raise the numerator and denominator to the same power. So say we have the fraction 𝑎 𝑏. And it’s to the 𝑐 power. We need to take 𝑎 to the 𝑐 power, the numerator, and 𝑏 to the 𝑐 power, the denominator. And here, we’ve done that for each of our fractions.
Now we need to simplify. Negative one squared is negative one times negative one, which gives us positive one. Three squared is three times three, which is nine. Negative one squared is again positive one. Two squared is two times two, which is four. Three cubed is three times three times three. Well, three times three is nine. And then, we take that times three. So nine times three is 27. So three cubed is equal to 27. And lastly, two cubed is two times two times two. Two times two is four. And then, four times two is eight. So two cubed is equal to eight. So we have six times one-ninth times one-fourth times twenty-seven eighths. We can change six to look like a fraction, six over one.
Now, we could multiply all the numerators together and then multiply all the denominators together and then simplify. However, we could simplify here. We look for numbers on the numerator that can simplify with numbers on the denominator, for example, nine and 27. Nine goes into itself once. And nine goes into 27 three times. So twenty-seven ninths reduces to three over one. Now, we have a six on the numerator and a four on the denominator and an eight on the denominator. So we can simplify either the six and the four together or the six and the eight together. It doesn’t matter.
Let’s look at the six and the four. Two can go into four. And two can also go into six. Two goes into four twice. And two goes into six three times. This is why we also could’ve simplified six and eight because two goes into six. And it goes into eight.
Now, be careful here. Do not be tempted to simplify the two and the eight. They’re both on the denominator. When simplifying, one has to be on the numerator and one has to be on the denominator. So on the numerator, we have three times one times one times three, which is nine. And on our denominator, we have one times one times two times eight. That’s 16. Now, if we simplified correctly, nine sixteenths shouldn’t reduce anymore. So are there any numbers that we can factor out of nine and 16? No, there’s not.
Therefore, our final answer will be nine sixteenths.