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How to Simplify an Expression with Parentheses & Exponents

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Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has been teaching math for over 9 years. Amy has worked with students at all levels from those with special needs to those that are gifted.

Mathematical expressions can utilize parentheses and exponents to notate calculations differently. See how the order of operations (PEMDAS) creates consistency when notating these expressions through a series of examples. Updated: 10/30/2021

A Mathematical Expression

It might surprise you to find out that we use math on an almost daily basis. Our whole world operates on math; just think about going shopping. What happens when you go to check out? You have your items like the latest video game, a board game, and perhaps some notebooks for school. What happens to these items? They get scanned, and what pops up on the cashier's screen? Why, the price of each item. Isn't this price a mathematical number? Yes, it is. And, what happens as the cashier scans in the rest of your items? They are added to your total cost. At the end, the cashier tells you how much you need to pay to take your items home.

What just happened? Math is what happened. Specifically, addition is what happened. As each item is scanned, it is added to the rest. While you don't see it on screen, there is actually a mathematical expression that is being calculated; that's just an expression with numbers and operations, such as addition, subtraction, multiplication, and/or division.

Say, for instance, the costs of your items are $3.49, $12.99, and $46.99. The mathematical expression that is calculated is 3.49 + 12.99 + 46.99. This, of course, is easy to calculate. But as you get more and more involved in math, you will come across mathematical expressions that also use parentheses and exponents, such as (4 + 5) * (2^2 + 3).

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  • 0:01 A Mathematical Expression
  • 1:48 Order of Operations
  • 5:13 Example 1
  • 5:36 Example 2
  • 7:07 Lesson Summary
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Order of Operations

When you see these kinds of mathematical expressions, you need to be extra aware of the order of operations. Remember that the order of operations tells you in what order to make your calculations. It tells you that you should calculate your parentheses first, then your exponents, then your multiplication and divisions, and then, finally, your addition and subtractions. Only by following your order of operations will you come up with the right answer. If you don't, you will come up with a totally different answer.

For example, if you calculate (4 + 5) * (2^2 + 3) from left to right, ignoring the order of operations, you will get (4 + 5) * (2^2 + 3) = 9 * (2^2 + 3) = 18^2 + 3 = 324 + 3 = 327 for your answer.

Let's see what we get if we follow the order of operations. We'll do the parentheses first. This means that we perform the operations inside the parentheses first. As we perform the operations inside each set of parentheses, we also follow the order of operations. So, inside each set of parentheses, we still will look for parentheses first, then exponents, then multiplication and division, and then, finally, addition and subtraction.

Inside our first set of parentheses, we just have addition, so we can go ahead and perform that. Now, we have 9 * (2^2 + 3). Inside the second set of parentheses, we see an exponent and addition. We perform the exponent before the addition. So, we square the 2. We get 9 * (4 + 3).

Now, we perform the addition inside this set of parentheses. We get 9 * 7. We are now done with our parentheses. Next, are there any exponents? No. What about multiplication and division? Yes, there is multiplication. We get 63.

There isn't any addition or subtraction left, so we're done. Our true answer is 63. Isn't that a big difference from 327? We would have been way off in our answer if we ignored the order of operations.

An easy way to remember the order of operations is to think of the phrase 'Please Excuse My Dear Aunt Sally,' or 'PEMDAS.' The P stands for parentheses, the E stands for exponents, the M stands for multiplication, the D stands for division, the A stands for addition, and the S stands for subtraction. You have all your operations in order.

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