Practice Problems
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 4xand 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: 9mand 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 - 2mnis 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.