Simplify the following expressions.
Part a) Four 𝑧 minus 𝑧 plus nine 𝑧.
Another way of saying “simplify” when we’re talking about algebraic expressions is to say “collect like terms.” What that means is just to gather all of the terms that are alike. In this first expression, every single term is a 𝑧, so all we need to do is add or subtract as necessary.
Remember, when addition and a subtraction occur in the same sum, we simply go from left to right. Four 𝑧 minus 𝑧 is three 𝑧. Careful though, a common mistake here is to think that the answer is four. In fact, what it’s saying is that we have four 𝑧s and then we take one of these 𝑧s away. So we must be left with three 𝑧. Then we add three 𝑧 to the nine 𝑧, which is 12𝑧. Four 𝑧 minus 𝑧 plus nine 𝑧 is 12𝑧.
Part b) 𝑝 multiplied by four multiplied by seven multiplied by 𝑞𝑟.
When multiplying terms, we can rewrite the expression somewhat. Multiplication is what we call commutative, which means that we can perform it in any order. We can rewrite this expression as four multiplied by seven multiplied by 𝑝 multiplied by 𝑞𝑟.
Once we have the numbers together, we can do the easy part of the multiplication. Four multiplied by seven is 28. We try to avoid using multiplication signs when we have algebraic expressions. So 𝑝 multiplied by 𝑞 multiplied by 𝑟 can simply be written as 𝑝𝑞𝑟. 𝑝 multiplied by four multiplied by seven multiplied by 𝑞𝑟 simplifies to 28𝑝𝑞𝑟.
Now a little note about mathematical convention, the number should always be at the front of the expression where possible. The letters though can go in any order, though we do tend to keep them alphabetical.
Part c) Two 𝑦 cubed minus 𝑦 cubed minus three 𝑦 cubed.
Just because each term has a power doesn’t really change anything. A common mistake here is to think that we need to cube the numbers in front of the letters. In fact, since indices are calculated before multiplication, the question is simply saying that we have two lots of 𝑦 cubed. Then we subtract one 𝑦 cubed. Then we subtract another three of these 𝑦 cubed.
Once again, we’ll just go from left to right. If we have two 𝑦 cubes and then we subtract one 𝑦 cubed, we’re left with one 𝑦 cubed. We don’t actually write the number one. We simply write that as 𝑦 cubed.
𝑦 cubed minus three 𝑦 cubed is a little trickier, and we can use a number line to help us work it out. Let’s imagine we’re subtracting three from one. To do this, we start at the number one. We move down the number line three spaces. That takes us to negative two. Since one minus three is equal to negative two, one 𝑦 cubed minus three 𝑦 cubed is equal to negative two 𝑦 cubed. And this means that our expression two 𝑦 cubed minus 𝑦 cubed minus three 𝑦 cubed is equal to negative two 𝑦 cubed.