When you write something down, itโs important that the people who read it will understand what you actually meant. But language can be ambiguous. If we say in English, โtwo times three squaredโ, do we mean that:
English is allowed to be ambiguous. But math needs to be unambiguous and mean the same thing for everyone who reads it. So for this reason, there are some rules that weโve all agreed to that control what a math expression really means. These rules are called the โorder of operationsโ, which we review here.
Consider the expression . There are two math operations here: at some point two things will be multiplied, and at some point something will be raised to a power. The result depends on which operation you decide to do first: If you multiply , and then square the result, you end with . If you square , and then multiply that result by , you end with . So if we want all people everywhere to interpret in the same way, then only one of these can be correct.
One tool that we have to clearly tell readers which thing to do first is a pair of grouping symbols, like parentheses and brackets. If you intend to do the multiplication first, then writing clearly tells your reader to do that. And if you intend to execute the power first, then writing clearly tells your reader to do that.
If we find , we have the area of one of the small square garden plots on the left. Then if we double that, we have , the area of the entire left garden plot.
But if we find , then first we are doubling . So we are getting the area of a large square garden plot whose sides are twice as long. We end up with the area of the entire garden plot on the right.
The grouping symbols tell us what to work on first. In this exercise, we have grouping symbols within grouping symbols, so any operation in there (the addition) should be executed first:
If math expressions used grouping symbols for every arithmetic operation, we wouldnโt need โorder of operationsโ. Some computer systems work that way, requiring the use of grouping symbols all the time. But it is more common to allow math expressions that donโt have grouping symbols everywhere, like . Should the addition be done first, or should the multiplication be done first? We have a set of rules the world has agreed to, known as the โorder of operationsโ. They tell us what to do first.
After all exponentiation is done, start doing multiplication, division, and negation. These things all have equal priority. If there are more than one of them in your expression, do these things in order from left to right as you would naturally read the expression.
After all other arithmetic is done, addition and subtraction are all that is left. They have equal priority. If there are more than one of them in your expression, do these things in order from left to right as you would naturally read the expression.
To help remember the order of operations, consider the acronym PEMDAS. You might use mnemonic devices to help remember this such as โPlease Excuse My Dear Aunt Sallyโ, โPeople Eat More Donuts After Schoolโ, etc.
. With this expression, we have addition and multiplication. The order of operations says the multiplication has higher priority, so do that first:
. According to the order of operations, the first thing to do is the division. After that, weโll apply the addition and subtraction working left to right:
. This expression only has subtraction and addition. These operations tie for priority (even though โAโ comes before โSโ in PEMDAS). So we work left to right to do them:
. This expression has only division and multiplication. Again, we have two operations that tie for priority, so we do them in order from left to right:
. Here we have addition inside parentheses, and an exponent of outside. We must do the arithmetic inside the parentheses first:
. This expression has multiplication and an exponent. There are parentheses, but no operation inside them. Parentheses used this way are just to make it clear that the and are separate numbers, not to be confused with the number . Exponentiation has the higher priority, so weโll do that part first, and then multiply:
There are several different ways to write multiplication. We can use the symbols ,, and to mean multiplication. We can also write two things right next to each other with no symbol in between them to mean multiplication. That is what is happening in Item f, where the is written right next to the with no symbol in between.
Using a symbol for multiplication is called โexplicit multiplicationโ and not writing any symbol at all is called โimplicit multiplicationโ. For this textbook, explicit and implicit multiplication have the same priority in the order of operations. However there are some conventions out in the real world where implicit multiplication has a higher priority in the order of operations than explicit multiplication. You may have seen memes with expressions like that play on how the real world has more than one convention for the order of operations.
With the expression , weโll simplify whatโs inside the parentheses according to the order of operations, and then take divided by that expression as our last step:
The expression has two sets of parentheses, so our first step will be to simplify whatโs inside each of those first according to the order of operations. Once weโve done that, weโll apply the exponent and then finally divide:
Absolute value bars group the expression inside just like a set of parentheses would. In the expression , the first thing to do is subtract since that is inside the absolute value bars.
A fraction bar can create two groups, one in the numerator and one in the denominator. With the expression , the first arithmetic to take care of is adding and subtracting . These two groups can be worked on separately in any order.
Each of these implied groupings also ask you to do something once the arithmetic on the inside is completed. Actually taking the absolute value or the square root, perhaps. Doing the division in the case of a fraction. Raising something to a power. But before doing those things, all of the arithmetic inside the groups should be take care of.
. The absolute value bars group the so we must do that subtraction first. Then we take the absolute value and continue:
It would be a mistake to subtract first, because that is multiplied by the . So subtracting would violate the order of operations.
. The radical is grouping , which must be simplified first. Then we take the square root and continue:
. We recognize that the fraction bar is creating two groups. We should simplify the numerator and denominator separately according to the order of operations, and proceed from there:
To simplify this expression, the first thing we want to recognize is the role of the main fraction bar, which groups the numerator and denominator. This implies weโll simplify the numerator and denominator separately according to the order of operations, and then reduce the fraction that results:
The first expression is different. There are two actions here: a negation and exponentiation. According to the order of operations, the exponentiation has higher priority, so we should do first.
You may find yourself needing to raise a negative number to a power, and using a calculator to do the work for you. If you do not understand the issue described above, then you may get incorrect results.
Compute the following. In each part, the first expression asks you to exponentiate and then negate the result. The second expression has a negative number raised to a power.
You might notice in Checkpoint 15 that and each have the same result, . Itโs true that the results are the same, but the two expressions say different things. With , you raise to a power first, then negate. With , you negate first, then raise to a power. Itโs like two different roads that happen to lead to the same place, which happens sometimes.
In this challenge, your job is to create expressions, using addition, subtraction, multiplication, and parentheses. You may use the numbers, , and in your expression, using each number only once. For example, you could make the expression: .
