In this lesson, we’ll practice simplifying a variety of algebraic expressions. We’ll use two key concepts, combining like terms and the distributive property, to help us simplify.
Practice Makes Perfect
If you’ve ever played a sport, you know the importance of practice. No track star just shows up at a meet and expects to run a record-breaking 100 meters without training. The more challenging the concept, the more practice you need. If you ever watch professional football, you know that even with an incredible amount of practice, athletes can still make mistakes sometimes; a receiver may forget a route or a punt returner may drop the ball.But we know we can get better at whatever we’re trying to accomplish with practice. What’s true in sports is also true in algebra.
Simplifying algebraic expressions can be as tricky as mastering a play in football. Fortunately, when we practice algebra, we’re unlikely to get knocked to the ground – unless you’re playing full-contact algebra. But let’s not do that here.
Simplifying Algebraic Expressions
Here we’re going to practice simplifying algebraic expressions. Simplifying algebraic expressions is more or less defined by its title.
It involves distributing terms across parentheses and combining like terms in order to make an expression simpler. By simpler, we usually mean shorter, or more condensed.Why is this useful? Algebraic expressions can get cumbersome with all their various bits and pieces. Think of it like food. What if every time you wanted a cookie, you had to ask for it by its parts – flour, sugar, butter, eggs, etc. – as opposed to just saying ‘I want a cookie’? That would be tedious, and it would interfere with your cookie eating. When we simplify an expression, we’re combining what we can so we’re just dealing with cookies, not the things that make up cookies.
There are two main skills involved in simplifying algebraic expressions. First, there’s combining like terms. This is the process of simplifying expressions by joining terms that have the same variable. So if you have x + 2x, you can combine them to get 3x. If you have 5x^2 + 3x^2 + 9x, you can only combine the 5x^2 and 3x^2 since they are the only terms that share the same exponent.
But we can still make that expression simpler by saying 8x^2 + 9x.Second, there’s the distributive property. This helpful law tells us that a(b + c) = (ab) + (ac). So, let’s say we have 7(x + 2y).
Since they’re not like terms, we can’t add that x and 2y. But the distributive property tells us we can distribute the 7 across the parentheses, giving us 7x + 14y.
Okay, time for some practice. Let’s start simple: 2y + 4y + 9. How can we simplify this? Well, we have two like terms: 2y and 4y.
Both of these terms have the same exponent, y. Let’s combine them to get 6y + 9. Can we go any further? No.
The 6y and 9 don’t share an exponent, so that’s as far as we can simplify this one.Here’s a good one: 9 + 3t – 5. In this one, all we can combine are the 9 and the -5. So our final expression is 4 + 3t. That’s it.
Those first two were a good warm-up. Let’s try a longer one. What if we have 3x^2 + 4x + x^2 + 2 + 11x? A good first step is to get like terms next to each other. What are our like terms? 4x and 11x both have an x. What about 3x^2 and x^2? They are like terms as well. If we move things around, we get 3x^2 + x^2 + 4x + 11x + 2. Now we just need to combine the like terms.
We add 3x^2 and x^2 to get 4x^2. Then we combine 4x and 11x to get 15x. So our simplified expression is 4x^2 + 15x + 2. That’s much better!Up to this point, we’ve only dealt with one variable. That’s kind of like flag football. Let’s jump to the NFL by using two: 9m + 8n + 3mn + 4m -2mn + n. It’s a little trickier with multiple variables, isn’t it? But let’s do the same thing we did before – moving like terms next to each other.
There are two terms with just one m: 9m and 4m. Then there are two with one n: 8n and n. What else? Those two with an mn? They’re like terms, too. So with a little shuffling, we have 9m + 4m + 8n + n + 3mn – 2mn. 9m + 4m is 13m.
8n + n is 9n. And 3mn – 2mn is just mn. That gives us 13m + 9n + mn.
Let’s do one with some serious exponent work: (5x^2y)^3. First, let’s handle that 5 cubed. That’s 125. And what do you do with an exponent raised to an exponent? You multiply them together.
So that x^2 to the third will be x^6. And the y will just become y^3. So our simplified expression is 125(x^6)(y^3).Okay, here’s one that involves the distributive property: 4ab + a(3b + b^2). Remember, we can distribute that a across the parentheses.
That gets us 4ab + 3ab + ab^2. And do we have any like terms? 4ab and 3ab. Combine those to get 7ab + ab^2. That’s as far as we can take this one. It’s like in a chocolate chip cookie – the flour, eggs and whatnot just become cookie dough, but the chocolate chips, represented here by the ab^2, are still chocolate chips. They’re just tastier when baked into cookies.I think we’re ready for a bigger challenge: 6x(x + 2y) + 3y(2x – y) + 4(x^2 + y^2).
Okay, lots to do here. First, note that nothing inside the parentheses can be simplified. They all involve addition or subtraction with different variables. So let’s use the distributive property and distribute the terms outside the parentheses. First, 6x * x is 6x^2 and 6x * 2y is 12xy. Next, 3y * 2x is 6xy and 3y * –y is -3y^2. Then, 4 * x^2 is 4x^2 and 4 * y^2 is 4y^2.
That gives us 6x^2 + 12xy + 6xy – 3y^2 + 4x^2 + 4y^2. Let’s do some shuffling and get 6x^2 + 4x^2 -3y^2 + 4y^2 + 12xy + 6xy. 6x^2 + 4x^2 is 10x^2. -3y^2 + 4y^2 is just positive y^2. Then, 12xy + 6xy is 18xy. Okay, that means our final, simplified expression is 10x^2 + y^2 + 18xy.
That’s much simpler than where we started! Cookie metaphor or not, I think we’ve earned a treat.
In this lesson, we practiced simplifying algebraic expressions. We became experts at combining like terms and utilizing the distributive property. With these methods, we made long, complicated expressions much simpler and easier to understand.
We also talked football and cookies, because all things should involve exercise and baked goods.
Once this lesson is completed, you should be able to confidently:
- Work through algebraic expressions
- Combine like terms
- Use the distributive property