Properties of Exponents

Exponents have certain properties that arise due to the properties of multiplication. We’ll derive the properties of exponents by looking for patterns in several examples. All the exponent properties hold true for any real numbers, but right now we will only use whole number exponents.  First, we will look at a few examples that leads to the Product Property.

For example, the notation [latex]5^{4}[/latex] can be expanded and written as [latex]5\cdot5\cdot5\cdot5[/latex], or [latex]625[/latex]. The exponent only applies to the number immediately to its left, unless there are parentheses. So, [latex]-5^{4}=-(5\cdot5\cdot5\cdot5)=-625[/latex] and [latex](-5)^{4}=(-5\cdot -5\cdot -5\cdot -5)=625[/latex].

What happens if we multiply two numbers in exponential form with the same base? Consider the expression [latex]{2}^{3}{2}^{4}[/latex]. Expanding each term, this can be rewritten as [latex]\left(2\cdot2\cdot2\right)\left(2\cdot2\cdot2\cdot2\right)[/latex] or [latex]2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2=2^{7}[/latex]. Notice that [latex]7[/latex] is the sum of the original two exponents, [latex]3[/latex] and [latex]4[/latex].

The base stayed the same and we added the exponents. This is an example of the Product Property for Exponents.

Caution! When we are reading mathematical rules, it is important to pay attention to the conditions on the rule.  For example, when using the product property we may only apply it when the terms being multiplied have the same base and the exponents are integers. Conditions on mathematical rules are often given before the rule is stated, as in this example it says “For any real number [latex]a[/latex] and any integers [latex]m[/latex] and [latex]n[/latex].”

Simplify Expressions Using the Power Property of Exponents

We will now further expand our capabilities with exponents. We will learn what to do when a term with a power is raised to another power.  Let’s see if we can discover a general property.

Let’s simplify [latex]\left(5^{2}\right)^{4}[/latex]. In this case, the base is [latex]5^2[/latex] and the exponent is [latex]4[/latex], so we multiply [latex]5^{2}[/latex] four times: [latex]\left(5^{2}\right)^{4}=5^{2}\cdot5^{2}\cdot5^{2}\cdot5^{2}=5^{8}[/latex] (using the Product Rule—add the exponents).

[latex]\left(5^{2}\right)^{4}[/latex] is a power of a power. It is the fourth power of [latex]5[/latex] to the second power, and we saw above that the answer is [latex]5^{8}[/latex]. Notice that the new exponent is the same as the product of the original exponents: [latex]2\cdot4=8[/latex].

So, [latex]\left(5^{2}\right)^{4}=5^{2\cdot4}=5^{8}[/latex] (which equals 390,625, if we do the multiplication).

This leads to another rule for exponents—the Power Property for Exponents. To simplify a power of a power, we multiply the exponents, keeping the base the same. For example, [latex]\left(2^{3}\right)^{5}=2^{3\cdot 5}=2^{15}[/latex].

An example with numbers helps to verify this property.

[latex]\begin{array}{ccc}\hfill {\left({5}^{2}\right)}^{3}& \stackrel{?}{=}& {5}^{2\cdot 3}\hfill \\ \hfill {\left(25\right)}^{3}& \stackrel{?}{=}& {5}^{6}\hfill \\ \hfill 15,625& =& 15,625\hfill \end{array}[/latex]

When simplifying any expression with exponents, we must be careful to correctly identify the base that is raised to each exponent.

When there is a product and an exponent we have to be careful to apply the exponent to the correct quantity. According to the order of operations, expressions in parentheses are simplified before exponents are applied. We’ll see how this works in the next example.

In the next example, examples 1 and 2 look similar, but we get different results.

Simplify Expressions Using the Product to a Power Property

We will now look at an expression containing a product that is raised to a power. Look for a pattern.

Simplify this expression.

   [latex]\left(2\cdot3\right)^{4}[/latex]     Base = [latex]\left(2\cdot3\right)[/latex]; exponent = [latex]4[/latex]

[latex]=\left(2\cdot 3\right)\left(2\cdot 3\right)\left(2\cdot 3\right)\left(2\cdot 3\right)[/latex]      The base gets multiplied 4 times.

[latex]=\left(2\cdot2\cdot2\cdot2\right)\left(3\cdot{3}\cdot{3}\cdot{3}\right)[/latex]     Regroup using the commutative and associative properties of multiplication.

[latex]=\left(2^{4}\right)\left(3^{4}\right)[/latex]     Rewrite using exponential notation.

[latex]=16\cdot 81[/latex]

Notice that the exponent is applied to each factor of [latex]2\cdot 3[/latex]. So, we can eliminate the middle steps.

[latex]\begin{array}{l}\left(2\cdot 3\right)^{4} = \left(2^{4}\right)\left(3^{4}\right)\text{, applying the }4\text{ to each factor, }2\text{ and }3\\\\\,\,\,\,\,\,\,\,\,\,\,\;\;\;\;=16\cdot81\end{array}[/latex]

The product of two or more numbers raised to a power is equal to the product of each number raised to the same power.

The exponent applies to each of the factors. This leads to the Product to a Power Property for Exponents.

An example with numbers helps to verify this property:

[latex]\begin{array}{ccc}\hfill {\left(2\cdot 3\right)}^{2}& \stackrel{?}{=}& {2}^{2}\cdot {3}^{2}\hfill \\ \hfill {6}^{2}& \stackrel{?}{=}& 4\cdot 9\hfill \\ \hfill 36& =& 36\hfill \end{array}[/latex]

We have developed the properties of exponents for multiplication: