Complete the blanks in this equation. Something multiplied by something
𝑚 to the fourth [𝑚𝑛 to the fourth] equals three 𝑚 to the forth 𝑛 to the eighth
minus 𝑚 squared 𝑛 to the eighth.
The distributive property is being used here. But it’s our job to figure out what exactly is
being distributed. Our missing value here was distributed across this subtraction problem to
produce three 𝑚 to the fourth 𝑛 to the eighth minus 𝑚 squared 𝑛 to the eighth.
Let’s focus our energy on these two pieces. How would we go from 𝑚 times 𝑛 to the fourth to
𝑚 squared 𝑛 to the eighth? To figure out what was being multiplied here, we can use division. We
divide 𝑚 squared 𝑛 to the eighth by 𝑚 times 𝑛 to the forth. When we do that, our 𝑚 squared
becomes 𝑚 to the first power, and our 𝑛 to the eighth becomes 𝑛 to the fourth power. This is the
value we’ll plug in here, 𝑚 times 𝑛 to the fourth. It’s the value being distributed across this
Now we need to focus on our last missing term. We need to know what 𝑚 times 𝑛 to the
fourth was multiplied by to produce three 𝑚 to the fourth 𝑛 to the eighth. We can use the same
process here. We can use division to find out what was being multiplied. We’ll divide three
𝑚 to the fourth 𝑛 to the eighth by 𝑚𝑛 to the fourth. The three can’t be divided, so we keep
that. Our 𝑚 to the fourth becomes 𝑚 cubed, and our 𝑛 to the eighth becomes 𝑛 to the fourth.
This means the final missing value would be three 𝑚 cubed 𝑛 to the fourth.
Our two missing pieces were 𝑚 times 𝑛 to the fourth, three times 𝑚 cubed times 𝑛 to the fourth.