Simplify three 𝑥𝑦 times four 𝑥
cubed 𝑦 squared.
We need to think about what’s
happening when we have a whole number and a list of variables behind it. We know that this means we’re
multiplying three times 𝑥 times 𝑦. And inside the parentheses, we’re
multiplying four times 𝑥 cubed times 𝑦 squared. What would it mean to multiply
these values together? What would it look like? Well, it would look something like
this: Three times 𝑥 times 𝑦 times four times 𝑥 cubed times 𝑦 squared. But if we wanted to simplify, we
can do some regrouping. We could group the whole numbers
together, three times four, and the 𝑥- terms together, 𝑥 times 𝑥 cubed, and then
the 𝑦-terms together, 𝑦 times 𝑦 squared.
Three times four is 12. We know that 𝑥 is equal to 𝑥 to
the first power. And we also know that 𝑥 to the 𝑎
power times 𝑥 to the 𝑏 power is equal to 𝑥 to the 𝑎 plus 𝑏 power. That means we need to add one plus
three to get 𝑥 to the fourth power. And we’ll do the same thing for 𝑦,
which will give us 𝑦 cubed. And we don’t need to have those
multiplication signs between them. We can write them all together as
12𝑥 to the fourth power 𝑦 cubed.
Now, these are quite a lot of
steps. You might be wondering if you have
to write this out every time. And you wouldn’t need to. This is just to show you where we
get this from. Usually, if I was going to solve
something like this, I would say that we need to multiply like terms. Three times four is 12 and 𝑥 to
the first power times 𝑥 cubed equals 𝑥 to the fourth power. And finally, 𝑦 to the first power
times 𝑦 squared is 𝑦 cubed. Both methods show the simplified
form 12𝑥 to the fourth 𝑦 cubed.