Calculate one-third to the fifth power times one-third to the second power all divided by one-third to the fourth power, giving your answer in its simplest form.
Here we can see we’re multiplying and dividing. And all of our numbers have powers. They have exponents. It’s important to remember, when multiplying like basis, we can add their exponents. And when dividing, we can subtract their exponents.
First, let’s work within the brackets. We have one-third to the fifth power times one-third to the second power. So that will be one-third to the five plus two power. And five plus two is seven. And now when dividing, so we can imagine these brackets aren’t here anymore, when dividing, we subtract our exponents. So we have one-third to the seven minus four power, which is equal to one-third cubed. So we need to cube the top and the bottom, the one and the three. One cubed is one and three cubed is 27, giving us a final answer: one 27th.
Now we also could’ve solved this by strictly expanding and not using any of the properties. So we could’ve taken each fraction and raise it to the designated power, like we’ve done here. And we get one over 243 times one-ninth divided by one over 81. When multiplying fractions, we multiply straight across. We multiply the numerators together. And we multiply the denominators together. And we have one over 2187 divided by one over 81. Now when dividing, we actually multiplied by the second fraction’s reciprocal. So we have 81 over 2187 which reduces to one 27th.
So once again, our answer, in its simplest form, will be one 27th.