Exponential Expressions

Repeated multiplication of one factor can be expressed in a shorthand notation known as an exponential expression.

For example, 3 · 3 · 3 · 3 can be written as $3^4$.

The repeated factor, in this case the 3, is called the base. The number of times the 3 appears as a factor in the product, in this expression the 4, is known as the exponent.

An exponential expression written in its long form, 3 · 3 · 3 · 3, is called the expanded form.

The work in this section will be confined to numeric bases; in later lessons, exponential expressions with variable bases will be developed. Exponential expressions that have numbers as bases can be simplified—that is, the multiplication can be carried out and a product determined.

• If a base is a positive number, then the product represented by the exponential expression will always be positive.
• If the base is negative, then the exponent determines whether the product is positive or negative.

Recognizing a negative base in an exponential expression can be tricky. The simplest way to distinguish a negative base is to remember that such a base is always contained in parentheses.

Examples of negative bases are:

${\left(-3\right)}^5$ or ${\left(-1\right)}^6$

An exponent only acts upon the base that immediately precedes it. Placing the negative number inside parentheses is the way of showing that an exponent is acting upon a negative base. So in expanded form,

${\left(-3\right)}^5=\left(-3\right)\left(-3\right)\left(-3\right)\left(-3\right)\left(-3\right)$ and

${\left(-1\right)}^6=\left(-1\right)\left(-1\right)\left(-1\right)\left(-1\right)\left(-1\right)\left(-1\right)$

Contrast these with an exponential expression such as the following: −72. Since the base is not contained in parentheses, this expression does not have a negative base. Instead, there are actually two operations occurring—first, the exponent of 2 acts upon the 7, then second, the opposite of the quantity is found. Thus,

$-7^2=-\left(7^2\right)=-\left(7·7\right)=-\left(49\right)=-49$

The rules for simplifying exponential expressions with negative bases are an extension of the rules for multiplying and dividing signed numbers. If a negative base is raised to an even exponent, the resulting product will be positive. If a negative base is raised to an odd exponent, the resulting product will be negative.

Click on the link below to see example 1_10 in a new window. When you are finished close the window to return to the lesson.

Example 1_10

${\left(-3\right)}^5$ and ${\left(-2\right)}^4$

 

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