Simplify the expression four 𝑥 squared 𝑦 cubed divided by 10𝑥 squared 𝑦 all divided by two 𝑦 to the fourth power divided by three 𝑥 to the fourth power.
To begin, let’s go ahead and rewrite this. This way, it’s a little bit easier to see that we’re dividing two fractions. And when dividing two fractions, we actually multiply by the second fraction’s reciprocal. So we change the division to multiplication. And then, we’re multiplying by the reciprocal. So we flip the second fraction. And when multiplying fractions, we multiply the numerators together and the denominators together. So we can cancel anything on the numerators with anything on the denominators. Two can go into itself once. And it can also go into four twice. These 𝑥 squares can cancel.
Before cancelling anymore, let’s see what we have left. On the numerator, we have two 𝑦 cubed three and 𝑥 to the fourth. So two times three gives us six. And we have 𝑦 cubed. And we also have 𝑥 to the fourth. On the denominator, we have 10. And we have 𝑦. And we have 𝑦 to the fourth. So 𝑦 times 𝑦 to the fourth is where we add their exponents. So 𝑦 would be 𝑦 to the first power. And one plus four would give us five. So the six tenths can reduce. So how come we didn’t catch that before? Well, two could’ve went into 10. And that’s where it would’ve simplified.
So back to where we were, six tenths, that does reduce. We can simplify them both by two, making three-fifths. And then we have 𝑦 to the third on the numerator and 𝑦 to fifth on the denominator. We can think of this two ways. When dividing with like bases, we subtract their exponents. And three minus five would be negative two. But when we have a negative exponent, if it’s on the numerator, we can move it to the denominator and make it positive, and vice versa. If it would’ve been negative on the denominator, we could’ve moved it to the numerator to make it positive.
Another way to think about this is if 𝑦 cubed is on the top, that means there are three 𝑦s on the top. And 𝑦 to the fifth is on the bottom. So there will be five 𝑦s on the bottom. Three of them would cancel. And there would be two left on the bottom. And then lastly, we have an 𝑥 to the fourth power. And there are no 𝑥s in the denominator. So we can’t simplify anymore. Therefore, three 𝑥 to the fourth power divided by five 𝑦 squared will be our final answer.