When we multiply a number by itself, we square it or raise it to a power of 2. For example, [latex]{4}^{2}=4\cdot 4=16[/latex]. We can raise any number to any power. In general, the exponential notation [latex]{a}^{n}[/latex] means that the number or variable [latex]a[/latex] is used as a factor [latex]n[/latex] times.

In this notation, [latex]{a}^{n}[/latex] is read as the nth power of [latex]a[/latex], where [latex]a[/latex] is called the base and [latex]n[/latex] is called the exponent. A term in exponential notation may be part of a mathematical expression, which is a combination of numbers and operations. For example, [latex]24+6\cdot \frac{2}{3}-{4}^{2}[/latex] is a mathematical expression.

To evaluate a mathematical expression, we perform the various operations. However, we do not perform them in any random order. We use the order of operations. This is a sequence of rules for evaluating such expressions.

Recall that in mathematics we use parentheses ( ), brackets [ ], and braces { } to group numbers and expressions so that anything appearing within the symbols is treated as a unit. Additionally, fraction bars, radicals, and absolute value bars are treated as grouping symbols. When evaluating a mathematical expression, begin by simplifying expressions within grouping symbols.

The next step is to address any exponents or radicals. Afterward, perform multiplication and division from left to right and finally addition and subtraction from left to right.

Let’s take a look at the expression provided.

There are no grouping symbols, so we move on to exponents or radicals. The number 4 is raised to a power of 2, so simplify [latex]{4}^{2}[/latex] as 16.

Therefore, [latex]24+6\cdot \dfrac{2}{3}-{4}^{2}=12[/latex].

For some complicated expressions, several passes through the order of operations will be needed. For instance, there may be a radical expression inside parentheses that must be simplified before the parentheses are evaluated. Following the order of operations ensures that anyone simplifying the same mathematical expression will get the same result.

Watch the following video for more examples of using the order of operations to simplify an expression.

Using Properties of Real Numbers

For some activities we perform, the order of certain operations does not matter, but the order of other operations does. For example, it does not make a difference if we put on the right shoe before the left or vice-versa. However, it does matter whether we put on shoes or socks first. The same thing is true for operations in mathematics.

Commutative Properties

The commutative property of addition states that numbers may be added in any order without affecting the sum.

We can better see this relationship when using real numbers.

Similarly, the commutative property of multiplication states that numbers may be multiplied in any order without affecting the product.

Again, consider an example with real numbers.

It is important to note that neither subtraction nor division is commutative. For example, [latex]17 - 5[/latex] is not the same as [latex]5 - 17[/latex]. Similarly, [latex]20\div 5\ne 5\div 20[/latex].

Associative Properties

The associative property of multiplication tells us that it does not matter how we group numbers when multiplying. We can move the grouping symbols to make the calculation easier, and the product remains the same.

Consider this example.

The associative property of addition tells us that numbers may be grouped differently without affecting the sum.

This property can be especially helpful when dealing with negative integers. Consider this example.

Are subtraction and division associative? Review these examples.

As we can see, neither subtraction nor division is associative.

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.

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

Note that 4 is outside the grouping symbols, so we distribute the 4 by multiplying it by 12, multiplying it by –7, and adding the products.

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.

Multiplication does not distribute over subtraction, and division distributes over neither addition nor subtraction.

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

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

Now, distribute [latex]-1[/latex] and simplify the result.

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.

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.

For example, we have [latex]\left(-6\right)+0=-6[/latex] and [latex]23\cdot 1=23[/latex]. There are no exceptions for these properties; they work for every real number, including 0 and 1.

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.

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

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 [latex]\frac{1}{a}[/latex], that, when multiplied by the original number, results in the multiplicative identity, 1.

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