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Simplifying with Nested Parentheses (page 2 of 3)

Parentheses inside of parentheses are called "nested" parentheses. The process of simplification works the same way as in the simpler examples on the previous page, but you do need to be a little more careful as you work your way through the grouping symbols.

• Simplify 4[x + 3(2x + 1)]

With nested parentheses like this, the safest plan is to work from the inside out. So I'll take the 3 through the inner parentheses first, before I even think about dealing with the 4 and the square brackets. I'll also simplify as much as I can as I go along. Note that I write each step out completely as I go:

4[x + 3(2x + 1)]
4[x + 3(2x) + 3(1)]

4[x + 6x + 3]

4[7x + 3]

4[7x] + 4[3]

28x + 12

By the way, there is no particular significance to brackets ("[" and "]") versus parentheses versus curly braces ("{" and "}"). Using the different grouping symbols is just a nice way of helping the user keep track of the different pairs of symbols. This is similar to working in an Excel spreadsheet, where the pairs of parentheses in a formula are color-coded, as you can see below:

This color-coding helps you see which ")" ("close-paren") goes with which "(" ("open-paren") in a formula. The different types of grouping symbols serve the same purpose in mathematics as do the colored parentheses in the spreadsheet. Copyright © Elizabeth Stapel 2003-2011 All Rights Reserved

FYI: The traditional sequence of grouping symbols, working from the inside out, is "parentheses", then "brackets", and then "braces"; then you repeat the sequence, as necessary. But this is not, to my knowledge, a rule; it's just a common convention.

• Simplify 9 – 3[x – (3x + 2)] + 4

I won't do anything with the "9 –" or the "+ 4" until I simplify inside the brackets and parentheses. I'll work from the inside out:

9 – 3[x – (3x + 2)] + 4
9 – 3[x1(3x + 2)] + 4
9 – 3[x – 1(3x) – 1(2)] + 4
9 – 3[x – 3x – 2] + 4

9 – 3[–2x – 2] + 4

9 – 3[–2x] – 3[–2] + 4

9 + 6x + 6 + 4

6x + 19

It is not required that you write out this many (or this few) steps. You should be careful to do one step at a time, though, writing things out completely and simplifying as you go. You should do as many steps as you need in order to consistently arrive at the correct answer.

• Simplify 5 + 2{ [3 + (2x – 1) + x] – 2}

I'll work carefully from the inside out:

5 + 2{ [3 + (2x – 1) + x] – 2}
5 + 2{ [3 + 2x – 1 + x] – 2}

5 + 2{ [2x + x + 3 – 1] – 2}

5 + 2{ [3x + 2] – 2}

5 + 2{3x + 2 – 2}

5 + 2{3x}

5 + 6x

6x + 5

You can use the Mathway widget below to practice simplifying with parentheses. Try the entered exercise, or type in your own exercise. Then click the "paper-airplane" button to compare your answer to Mathway's. (Or skip the widget and continue with the lesson.)

(Clicking on "Tap to view steps" on the widget's answer screen will take you to the Mathway site for a paid upgrade.)

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 Cite this article as: Stapel, Elizabeth. "Simplifying with Nested Parentheses." Purplemath. Available from     http://www.purplemath.com/modules/simparen2.htm. Accessed [Date] [Month] 2016

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