# Mathematical Parentheses

Hi, and welcome to this video about using parentheses in math! You should be familiar with how parentheses work in writing. When we’re writing, we can use parentheses to add extra information to an idea in a sentence. Typically, if the information in parentheses was removed, the meaning of the sentence would not change. However, in math, the use of parentheses within an expression is necessary to group terms and to define the order of algebraic operations. If parentheses are removed, it can drastically change the mathematical relationships and, therefore, the result of your equation.

Today, we’ll be looking at the different ways parentheses are used in math and why the placement of parentheses is so important.

First, let’s review a couple of the basics. The **order of operations**, or PEMDAS, is something we use to evaluate an expression or to solve an equation. The P in PEMDAS stands for “parentheses,” which means that whatever is contained in the parentheses should be addressed first in the process.

Now, before we get into how we can use parentheses, let’s remember what an algebraic expression actually is. An algebraic expression is made up of terms “joined” by the operations of addition or subtraction. A **term** can be a constant, a **variable**, or the product of two or more variables, and the variables can even be raised to powers. When a term has a constant multiplied by a variable, the constant is called a **coefficient**.

When we evaluate these expressions, that means we are substituting particular values for each variable. Parentheses should be used to contain the value in the expression so the correct order of operations is followed.

Consider the following example:

\(2x^{2}+3x-4\text{ if }x=-1\)

Substituting the value of \(-1\) for the variable \(x\) in this expression is pretty straightforward. That gives us \(2(-1)^{2}+3(-1)-4\).

We have placed the \(-1\)s in parentheses to contain their value. If we left them out of the parentheses, we would just have \(2-1^{2}\) and \(3-1-4\). Instead, we have \(2\times (-1)^{2}\) and \(3\times (-1)-4\). Let’s work through it!

Now, because the exponent, 2, is outside of the parentheses, we’re going to square the value contained in the parentheses first, then multiply the result by our coefficient, 2. The square of negative 1 is positive 1, which gives us \(2\cdot 1+3(-1)-4\).

Now we’re going to simplify even further by multiplying out our numbers. We have \(2-3-4\). And if we simplify even further we get \(2-3=-1\), \(-1-4=-5\).

In contrast, let’s look at the same expression but without the parentheses. So we have: \(2-1^{2}+3-1-4\).

Since there are no parentheses, we have to start with the exponent. 1 squared is just 1, so we’re left with \(2-1+3-1-4\), which if you add those all up, you get -1. As you can see, adding parentheses can really change the outcome of our expression!

Another way we use parentheses in math is for substitution. One method of solving a system of equations requires you to substitute an expression into an equation to solve for a variable. Let’s look at an example:

Solve the system of equations:

\(x = 2y – 3\) and \(3x – 7y = -14\)

Since this first equation tells us the value of \(x\), we can actually substitute this for \(x\) in the second equation using parentheses.

So now, instead of \(3x-7y\), we have \(3(2y – 3) – 7y = -14\).

As you can see, all we have to do now is solve this equation for \(y\). Because the expression in the parentheses is already simplified and it is not being raised to a power, the next step in the order of operations is multiplication. The coefficient of 3 in front of the parentheses must be distributed into the parentheses by multiplication. That means that we need to multiply each term by 3:

\(3\times 2y=6y\); \(3\times -3=-9\)

\(6y – 9 – 7y = -14\)

Now, we need to combine like terms, leaving the \(y\)-variable on the left side of the equation.

\(6y-7y=-1y\)

\(-1y – 9 = -14\)

Solving for \(y\) is now possible by adding 9 to both sides of the equation and then multiplying both sides by -1.

\((-1y – 9)-9 = -14 + 9\)

\((-1)(-1y)=-5(-1)\)

\(y = 5\)

The final step to solve this system is to substitute \(y=5\) into the equation \(x=2y-3\) and solve for \(x\).

\(x = 2(5) – 3=10-3=7\)

The solution to the system is the ordered pair, \((7,5)\).

Now that we’ve got the hang of using parentheses, let’s look at an expression that has a parentheses inside parentheses!

\((5 + 2)(3 – (-14) \div -2)\)

This may appear overwhelming, but remember that parentheses must be dealt with first in the evaluation process. The first set of parentheses simplifies to 7, but be careful on the second set. There is both subtraction and division within the parentheses, and PEMDAS dictates that division takes precedence over subtraction, so perform the division of \(-14\div -2\) first for an answer of positive 7. So we have \(7(3-7)\).

The final set of parentheses simplifies quickly to -4. And then multiplying, \(7\times -4=-28\).

Now, it is important to remember that inputting parentheses needs to be done carefully when entering your expressions and functions into your calculator. As discussed before, \(-3^{2}\) returns a different value than \((-3)^{2}\).

For more complex work on a graphing calculator, the use of parentheses is essential to define a function. For example, suppose you are graphing this rational expression: \(\frac{x^2+5}{x+2}\).

If you do not use parentheses to define the numerator and denominator in your calculator, it would read this expression as \(x^{2}+5\div x+2\), as opposed to the intended rational function \(\frac{x^2+5}{x+2}\).

As you can see, parentheses have several different uses in math problems, all of which are aimed at making it easier to solve those problems.

I hope this review was helpful! Thanks for watching, and happy studying!

## Practice Questions

**Question #1:**

Simplify the following expression using PEMDAS:

\(31+(2^2+14)^2-6\)

349

318

315

311

**Answer:**

The correct answer is 349. Remember, PEMDAS is the set of rules that prioritizes calculations when simplifying an expression or equation. For the expression \(31+(2^2+14)^2-6\), first simplify the value inside the parentheses including the exponents. \((2^2+14)\) becomes \((18)\). Now, address the exponents outside of the parentheses. \(31+(18)^2-6\) becomes \(31+324-6\), which simplifies to 349.

**Question #2:**

Simplify the following expression using PEMDAS:

\(4^3+2(3^2-1)+2^2\)

63

184

84

76

**Answer:**

The correct answer is 84. Use PEMDAS to simplify the expression. For the expression \(4^3+2(3^2-1)+2^2\), first simplify the value inside the parentheses including the exponent. \((3^2-1)\) simplifies to \((8)\). The expression can then be rewritten as \(4^3+2(8)+2^2\). The next step is to simplify the exponents. Rewrite \(4^3\) as 64, and \(2^2\) as 4. The expression is now written as \(64+2(8)+4\). The next step is to multiply. Multiply \(2(8)\) to get 16. Now the expression is written as \(64+16+4\), which simplifies to 84.

**Question #3:**

Simplify the following expression if \(x=3\).

\(4(5x-2)-6x\)

46

11

34

74

**Answer:**

The correct answer is 34. Once again, use PEMDAS to simplify the expression. Because x=3, plug in 3 for every x in the expression. The expression is now written as \(4(5(3)-2)-6(3)\). Before moving further, work through the math that is inside the parentheses, \((5(3)-2)\) by multiplying 5 times 3 before subtracting. This reduces to (13) , so now the expression reads \(4(13)-6(3)\). Next, move on to multiplication. Simplify \(4(13)\) and \(6(3)\), which would be \(52-18\). This simplifies to 34 for the final answer.

**Question #4:**

Simplify the following expression if \(a=2\) and \(b=-4\).

\(6^2+5(a+b)+a^2+4(b)\)

32

11

45

14

**Answer:**

The correct answer is 14. Once again, use PEMDAS to simplify the expression. Because \(a=2\) and \(b=-4\), plug in these values for each variable. Now the expression is written as \(6^2+5(2+(-4))+2^2+4(-4)\). The next step is to address what’s inside the parentheses. \((2+(-4)\) simplifies to \((-2)\), which means the expression is now written as \(6^2+5(-2)+2^2+4(-4)\). Next is dealing with the exponents. After simplifying the exponents, the expression is now written as \(36+5(-2)+4+4(-4)\). Next is multiplication. Multiply \(5(-2)\) to get \(-10\), and multiply \(4(-4)\) to get \(-16\). The expression now reads \(36-10+4-16\). From here, simply add and subtract from left to right to get a final answer of 14.

**Question #5:**

Simplify the following expression if \(x=5\).

\((2x)^2+4(3x-2)-16\)

136

146

121

164

**Answer:**

The correct answer is 136. Once again, use PEMDAS to simplify the expression. Because \(x=5\), plug in 5 for every *x* in the expression. The expression is now written as \((2(5))^2+4(3(5)-2)-16\). Address the math inside the parentheses first. Simplify \((2(5))\) to \(10\), and simplify \((3(5)-2)\) to \(13\). Now the expression reads as \((10)^2+4(13)-16\). Now move on to the exponents. Simplify \((10)^2\) to \(100\) so that the new expression reads as \(100+4(13)-16\). Next is multiplication. Multiply \(4(13)\) to get \(52\). The expression is now written as \(100+52-16\), which simplifies to \(136\).