The correct use of parentheses is a basic math skill which many students haven't yet acquired. This skill is crucial on the SAT and ACT. Understanding how to use parentheses correctly can spare you stress and pave the way to answering questions correctly.
You may not have to learn higher-level math in order to succeed on the SAT / ACT; there are tricks and shortcuts for many of the questions. But a certain groundwork of basic algebra skills is necessary if you wish the tricks to work. Of all the skills which comprise this groundwork, the proper use of parentheses is the most important.
Today we'll look at a series of algebraic expressions. For each expression, we'll look at it in two different forms: one with a set of parentheses, and one without. By seeing the difference between them, hopefully you'll understand exactly how parentheses change a mathematical expression.
The order of operations
Perhaps you've heard of the acronym PEMDAS:
MD Multiplication / Division
AS Addition / Subtraction
PEMDAS represents the order of operations. In the natural order of mathematical operations, exponents trump all else, multiplication and division follow, and only then does addition or subtraction occur.
The role of parentheses in the order of mathematical operations
Parentheses play two main roles in mathematical expressions. They are as follows:
1) Manipulation of the order of operations
2) Distribution of a single operation among many elements.
We'll first address role #1, the manipulation of the order of operations. Let's start with the following two expressions:
1) 5 · 3 + 7²
2) 5 · (3 + 7)²
The value of the first expression is 64. The value of the second expression is 500.
The difference lies in the parentheses. The first equation utilizes no parentheses, which means that the order of operations is as normal. First we take the exponent on the 7, yielding 49, and then add this to 15 to get 64.
Since the second expression has parentheses, we first tackle whatever's inside them. The only expression within the parentheses is a sum: 3 + 7 = 10. This is the number we'll now raise by an exponent, yielding 100.
That is, the addition in this equation preceded the exponent, even though it's not supposed to by the common order of operations.
100 is then multiplied by 5, giving us a final answer of 500.
The distribution of operations
Parentheses are sometimes used to indicate that a single operation should be implemented upon a number of elements.
1) 3*(x + 4 + 2y) = 3x + 12 + 6y
2) 4 - (2 + x) = 4 - 2 - x = 2 - x
3) (x + 6) / 2 = x / 2 + 3
4) (4y)² = 4² * y² = 16 * y²
In each of the examples above, an operation is implemented on everything within the parentheses.
In Example #1, 3 is multiplied by everything.
In Example #2, everything within the parentheses is subtracted.
In Example #3, everything within the parentheses is divided by 2.
In Example #4, everything in the parentheses is raised to the second power.
More on this later in the article.
Always address parentheses first
When simplying a mathematical operation, the first step is to get rid of parentheses. Distribute whatever operation needs to be distributed, or simplify the expression inside the parentheses first. The next time you write the expression, parentheses shouldn't be present. It's difficult and confusing to work with any mathematical expression which still has a pair of parentheses in it.
Common parentheses mistakes
The truth is, most students can deliver the correct answer when shown mathematical expressions like the ones above. They intuitively know what to do, at least with simple expressions. But the situation becomes stickier when they have to write the expression themselves (like in word problems) or when variables manifest. Today we'll look at some mistakes students make with parentheses on the SAT / ACT, and learn the best ways to correct them.
Subtracting a set of elements
Let's look at two different expressions, per our usual arrangement.
1) 7 - 3x + 4y - 19
2) 7 - (3x + 4y - 19)
In Expression 1), only the 7 and -19 can be added. The most simplified version is -12 - 3x + 4y.
Now let's look at Expression 2). Remember the second purpose of parentheses: to distribute the operation amongst every element in the parentheses.
In Expression 2), the minus sign has to be distributed to each of the elements within the parentheses. Remember that the first step in simplifying an algebraic expression is always to simplify the inside of the parentheses, and then to get rid of the parentheses. We'll do that here:
7 - (3x + 4y - 19) = 7 - 3x - 4y + 19 = 26 - 3x - 4y
Note that this is a different result than that of Expression 1). Not only is the numeric value different (26 as opposed to 12), but the sign of 4y is now minus.
This is the source of many mistakes students make on the SAT / ACT. You must put parentheses around the entire expression you wish to subtract.
A word of caution: when interpreting word problems where an algebraic expression, as opposed to a single element, has to be subtracted, students sometimes feel tempted to do away with parentheses. They assume they can subtract it correctly.
Please don't do that. Unless you really excel in these matters, subtracting a slew of elements is a surefire way to make mistakes. Write a subtraction sign, open parentheses, write the slew of things that need to be subtracted, and close parentheses. Only then should you begin flipping the sign of each element.
Multiplying negative numbers
This is not technically necessary, but it's advisable in order to avoid confusion. Almost inevitably, students who abstain from parentheses when multiplying negative numbers end up miscalculating.
Let's say I want to multiply the three numbers -2, -5, and -3. There are two main ways I could write this:
1) -2 · -5 · -3
2) (-2) · (-5) · (-3)
Do you notice how the negatives in Expression 1) look a lot like subtraction signs? And this is when everything is neatly typed. If you're writing the expression by hand, the possibility of error is immense. Do yourself a favor and use parentheses, à la Expression 2).
Exponents and Parentheses
There's one common principle that accompanies exponents. It's this:
If there's a set of parentheses immediately before an exponent, everything in the parentheses is raised to that power.
If there is no set of parentheses immediately before the exponent, only the last number / variable is raised to that power.
There are a number of mistakes students make when using parentheses with exponents. We'll review the most common ones now.
Mistake #1: Raising a product to an exponent
Let's look at an expression that students usually get right:
(3 · 4)²
Due to the parentheses, the exponent doesn't take precedence. We'd first multiply what's inside the parentheses. Only then would we raise the expression inside the parentheses - 12 - to the exponent of 2, getting a final answer of 144.
You could solve this in two ways, according to the two mathematical purposes of parentheses. The first purpose of parentheses is to establish precedence. For that reason, we'd first multiply what's inside the parentheses, and only then raise it to an exponent.
3 · 4 = 12, and 12² = 144. The final answer would be 144.
You could also approach this through the second purpose of parentheses, which is to distribute an operation to everything inside the parentheses. In this case, that means the exponent of 2 would be true of both the 3 and the 4:
(3 · 4)² = 3² · 4² = 9 · 16 = 144
Let's try a similar example, this one with a variable:
(3 · x)²
This expression is the counterpart of the one before it, except that here you can't simplify what's inside the parentheses. The 3x will remain 3x.
Since 3x can't be simplified further, it's easier to solve through the second purpose of parentheses: distributing an operation. That is,
(3 · x)² = 3 ² · x² = 9 · x²
A common mistake is attaching the exponent only to the last element, instead of to the whole thing:
(3 · x)² ≠ 3x²
When dealing with an exponent which isn't just upon a single number or variable, remember parentheses.
Mistake #2: Negative base in an exponent
Now we'll discuss another parentheses mistakes commonly made with exponents. Let's look at two expressions and see the difference between them.
2) (-3)² The first expression has no parentheses, so it follows the usual order of operations. First take care of the exponent:
-3² = -9
Let's look at the second expression.
Since there are parentheses immediately before the exponent, everything within the parentheses is being raised to the second power. That includes the 3 and the negative.
(-3) * (-3) = 9
Note that we got two different results for the two expressions.
Never forget parentheses when the things being multiplied are negative.
When are parentheses unnecessary?
If you understand when parentheses don't change the expression, you likely understand when parentheses do change the expression. The following are the cases in which parentheses don't make a difference.
Case #1: Parentheses around the entire expression
(3x + 4*7 - x³)
The parentheses above do nothing. Remember that the purpose of parentheses is either to change the order of operations, or to distribute an operation among everything within the parentheses.
The way parentheses change order of operations is by first taking care of the elements within the parentheses, and only then moving on to the things outside. Here, everything is within the parentheses, which means that there's nothing to prioritize. Similarly, there's no operation outside the parentheses to distribute, since everything is within. Parentheses would be useless.
Case #2: parentheses around a single number or variable.
Again, the purpose of parentheses is either to change the order of operations, or to distribute an operation among everybody within the parentheses.
If your parentheses are around a single number or variable, there's nothing to take precedence in terms of orders of operation. This is because there are no operations within the parentheses - it's just one number or variable. There's also nothing to distribute among the various elements in the parentheses, since there's only one element in the parentheses. The operator would have been implemented upon it anyway.
Case #3: When the operator is addition
When the operator implemented upon the parentheses is addition, parentheses are unnecessary.
Let's look at the following expression:
y² - 4x + (3 + 2y -3x)
The first step is always to take care of parentheses. There are no like terms, so we'll just simplify:
y² - 4x + (3 + 2y -3x) = y² - 4x + 3 + 2y -3x
Note that the expression with the parentheses and the expression without them look exactly the same.
Remember that the purpose of parentheses is to change the order of operation, such that the operations inside the parentheses would come before the operations outside it. But addition is last in order of operations anyway, which means that parentheses wouldn't have changed anything.
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