## Use Properties of Real Numbers

### Learning Outcomes

• Simplify expressions with real numbers that require all operations

## Order of Operations

You may or may not recall the order of operations for applying several mathematical operations to one expression. Just as it is a social convention for us to drive on the right-hand side of the road, the order of operations is a set of conventions used to provide order when you are required to use several mathematical operations for one expression. The graphic below depicts the order in which mathematical operations are performed. Order of operations

Order of operations are a set of conventions used to provide a formulaic outline to follow when you are required to use several mathematical operations for one expression.  The box below is a summary of the order of operations depicted in the graphic above.

### The Order of Operations

• Perform all operations within grouping symbols first. Grouping symbols include parentheses ( ), brackets [ ], braces { }, and fraction bars.
• Evaluate exponents or square roots.
• Multiply or divide, from left to right.
• Add or subtract, from left to right.

This order of operations is true for all real numbers.

### Example

Simplify $7–5+3\cdot8$

In the following example, you will be shown how to simplify an expression that contains both multiplication and subtraction using the order of operations.

When you are applying the order of operations to expressions that contain fractions, decimals, and negative numbers, you will need to recall how to do these computations as well.

### Example

Simplify $3\cdot\dfrac{1}{3}\normalsize -8\div\dfrac{1}{4}$

### Try It

If the expression has exponents or square roots, they are to be performed after parentheses and other grouping symbols have been simplified and before any multiplication, division, subtraction, and addition that are outside the parentheses or other grouping symbols.

## Exponents

When you are evaluating expressions, you will sometimes see exponents used to represent repeated multiplication. Recall that an expression such as $7^{2}$ is exponential notation for $7\cdot7$. (Exponential notation has two parts: the base and the exponent or the power. In $7^{2}$, $7$ is the base and $2$ is the exponent; the exponent determines how many times the base is multiplied by itself.)

Exponents are a way to represent repeated multiplication; the order of operations places it before any other multiplication, division, subtraction, and addition is performed.

### Example

Simplify $3^{2}\cdot2^{3}$.

In the video that follows, an expression with exponents on its terms is simplified using the order of operations.

## Grouping Symbols

Grouping symbols such as parentheses ( ), brackets [ ], braces$\displaystyle \left\{ {} \right\}$, fraction bars, and roots can be used to further control the order of the four arithmetic operations. The rules of the order of operations require computation within grouping symbols to be completed first, even if you are adding or subtracting within the grouping symbols and you have multiplication outside the grouping symbols. After computing within the grouping symbols, divide or multiply from left to right and then subtract or add from left to right. When there are grouping symbols within grouping symbols, calculate from the inside to the outside. That is, begin simplifying within the innermost grouping symbols first.

Remember that parentheses can also be used to show multiplication. In the example that follows, both uses of parentheses—as a way to represent a group, as well as a way to express multiplication—are shown.

### Example

Simplify $\left(3+4\right)^{2}+\left(8\right)\left(4\right)$

### Example

Simplify  $4\cdot{\frac{3[5+{(2 + 3)}^2]}{2}}$

### Try It

In the following video, you are shown how to use the order of operations to simplify an expression with grouping symbols, exponents, multiplication, and addition.

### Try it

Square roots are another grouping symbol.  Operations inside of a square root need to be performed first. In the next example, we will simplify an expression that has a square root.

### Example

Simplify $\dfrac{\sqrt{7+2}+2^2}{(8)(4)-11}$

These problems are very similar to the examples given above. How are they different and what tools do you need to simplify them?

a) Simplify $\left(1.5+3.5\right)–2\left(0.5\cdot6\right)^{2}$. This problem has parentheses, exponents, multiplication, subtraction, and addition in it, as well as decimals instead of integers.

Use the box below to write down a few thoughts about how you would simplify this expression with decimals and grouping symbols.

b) Simplify ${{\left(\dfrac{1}{2}\normalsize\right)}^{2}}+{{\left(\dfrac{1}{4}\normalsize\right)}^{3}}\cdot \,32$

Use the box below to write down a few thoughts about how you would simplify this expression with fractions and grouping symbols.

## Combining Like Terms

One way we can simplify expressions is to combine like terms. Like terms are terms where the variables match exactly (exponents included). Examples of like terms would be $5xy$ and $-3xy$ or $8a^2b$ and $a^2b$ or $-3$ and $8$.  If we have like terms we are allowed to add (or subtract) the numbers in front of the variables, then keep the variables the same. As we combine like terms we need to interpret subtraction signs as part of the following term. This means if we see a subtraction sign, we treat the following term like a negative term. The sign always stays with the term.

This is shown in the following examples:

### Example

Combine like terms:  $5x-2y-8x+7y$

In the following video you will be shown how to combine like terms using the idea of the distributive property.  Note that this is a different method than is shown in the written examples on this page, but it obtains the same result.

### Example

Combine like terms:  $x^2-3x+9-5x^2+3x-1$

In the video that follows, you will be shown another example of combining like terms.  Pay attention to why you are not able to combine all three terms in the example.

### Distributive Property

The distributive property states that the product of a factor times a sum is the sum of the factor times each term in the sum.

$a\cdot \left(b+c\right)=a\cdot b+a\cdot c$

This property combines both addition and multiplication (and is the only property to do so). Let us consider an example.

### Example

Use the distributive property to show that $4\cdot[12+(-7)]=20$

To be more precise when describing this property, we say that multiplication distributes over addition.

The reverse is not true as we can see in this example.

$\begin{array}{ccc}\hfill 6+\left(3\cdot 5\right)& \stackrel{?}{=}& \left(6+3\right)\cdot \left(6+5\right) \\ \hfill 6+\left(15\right)& \stackrel{?}{=}& \left(9\right)\cdot \left(11\right)\hfill \\ \hfill 21& \ne & \text{ }99\hfill \end{array}$

A special case of the distributive property occurs when a sum of terms is subtracted.

$a-b=a+\left(-b\right)$

For example, consider the difference $12-\left(5+3\right)$. We can rewrite the difference of the two terms $12$ and $\left(5+3\right)$ by turning the subtraction expression into addition of the opposite. So instead of subtracting $\left(5+3\right)$, we add the opposite.

$12+\left(-1\right)\cdot \left(5+3\right)$

Now, distribute $-1$ and simplify the result.

$\begin{array}{l}12-\left(5+3\right)=12+\left(-1\right)\cdot\left(5+3\right)\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=12+[\left(-1\right)\cdot5+\left(-1\right)\cdot3]\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=12+\left(-8\right)\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=4\end{array}$

### Example

Rewrite the last example by changing the sign of each term and adding the results.

This seems like a lot of trouble for a simple sum, but it illustrates a powerful result that will be useful once we introduce algebraic terms.

### Identity Properties

The identity property of addition states that there is a unique number, called the additive identity (0) that, when added to a number, results in the original number.

$a+0=a$

The identity property of multiplication states that there is a unique number, called the multiplicative identity (1) that, when multiplied by a number, results in the original number.

$a\cdot 1=a$

### Example

Show that the identity property of addition and multiplication are true for $-6 \text{ and }23$.

Inverse Properties

The inverse property of addition states that, for every real number a, there is a unique number, called the additive inverse (or opposite), denoted a, that, when added to the original number, results in the additive identity, $0$.

$a+\left(-a\right)=0$

For example, if $a=-8$, the additive inverse is $8$, since $\left(-8\right)+8=0$.

The inverse property of multiplication holds for all real numbers except $0$ because the reciprocal of $0$ is not defined. The property states that, for every real number a, there is a unique number, called the multiplicative inverse (or reciprocal), denoted $\dfrac{1}{a}$, that, when multiplied by the original number, results in the multiplicative identity, $1$.

$a\cdot\dfrac{1}{a}\normalsize =1$

### Example

1) Define the additive inverse of $a=-8$, and use it to illustrate the inverse property of addition.

2) Write the reciprocal of $a=-\dfrac{2}{3}$, and use it to illustrate the inverse property of multiplication.

### A General Note: Properties of Real Numbers

The following properties hold for real numbers a, b, and c.

Commutative Property $a+b=b+a$ $a\cdot b=b\cdot a$
Associative Property $a+\left(b+c\right)=\left(a+b\right)+c$ $a\left(bc\right)=\left(ab\right)c$
Distributive Property $a\cdot \left(b+c\right)=a\cdot b+a\cdot c$
Identity Property There exists a unique real number called the additive identity, 0, such that, for any real number a

$a+0=a$
There exists a unique real number called the multiplicative identity, 1, such that, for any real number a

$a\cdot 1=a$
Inverse Property Every real number a has an additive inverse, or opposite, denoted $–a$, such that

$a+\left(-a\right)=0$
Every nonzero real number a has a multiplicative inverse, or reciprocal, denoted $\dfrac{1}{a}$, such that

$a\cdot \left(\dfrac{1}{a}\normalsize\right)=1$

### Example

Use the properties of real numbers to rewrite and simplify each expression. State which properties apply.

1. $3\left(6+4\right)$
2. $\left(5+8\right)+\left(-8\right)$
3. $6-\left(15+9\right)$
4. $\dfrac{4}{7}\normalsize\cdot \left(\dfrac{2}{3}\normalsize\cdot\dfrac{7}{4}\normalsize\right)$
5. $100\cdot \left[0.75+\left(-2.38\right)\right]$

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