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How do I simplify?

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2 Answers

Jeanna, when we simplify an expression like this with parentheses, brackets, and braces, we begin by removing the parentheses using the Distributive Property.  The parentheses are inside the brackets and the brackets are inside the braces.

We have 3[(5c - 25 + 15) - (10c - 6 + 4)].

Notice the braces are no longer needed and have been replaced with brackets and the brackets in the original expression are now parentheses.  We simplify now inside the parentheses.

We have 3[(5c -10) - (10c - 2)].

Now we can remove the parentheses and simplify inside the brackets.

We have 3(5c - 10 - 10c + 2) and then 3(-5c - 8).

We multiply using the Distributive Property and the simplified expression is -15c -24.





The main thing to keep in mind here is Order of Operations, also commonly referred to as "PEMDAS." The letters describe the order things should be done in: Parenthesis, Exponents, Multipication and Division, Addition and Subtraction. Within this rule, you work from left to right.

In the problem shown, you have 3 layers of parenthesis that make this thing look like a beast. It's not so scary once you start working on it, though. The key is to work from the inside, to the outside. That means we want to try simplifying the (c-5) and the (5c-3) first.

In this case, you can't really do anything with them, since you don't know what c is. So you move on to [5(c-5)+15] and [2(5c-3)+4]. Because a number or letter next to a parenthesis counts as multiplication, we'll do that first, so let's ignore the +15 and +4 for the moment. To work with 5(c-5) and 2(5c-3), you'll need to use the distributive property.

5(c-5) = 5c-25
2(5c-3) = 10c-6

Now the equation is 3{[5c-25+15]-[10c-6+4]}. The 5c and 10c still can't be added to anything, but we can work with the numbers that follow. The negative numbers can make things tricky, though. The best thing to do is to pretend you are adding a negative number, so the subtraction isn't lost. (You can change it back when you're done, if it makes more sense.)

[5c-25+15] = 5c+(-25)+15 = 5c+(-10) = 5c-10
[10c-6+4] = 10c+(-6)+4 = 10c+(-2) = 10c-2

Now we have 3{[5c-10]-[10c-2]}. Once again, there is sign-swapping involved. Whenever swapping signs, the key is to make sure you do it evenly. The reason this works is that if you are subtracting a negative, you are actually adding a positive. For example, 2-(-2) = 2+2.

[5c-10]-[10c-2] = [5c-10]+[-10c+2] = 5c-10c+(-10)+2 = -5c+(-8) = -5c-8

For the final piece, you'll be using the distributive property again. Also, keep in mind that a negative times a positive is always a negative. (On the other hand, positive times positive and negative times negative will both always be positive.)

3{-5c-8} = 3(-5c)+3(-8) = -15c+(-24) = -15c-24




Note: Sorry, I initially typed in that 3*8 = 28, when 3*8 = 24. I just noticed my typo.