Exponents

Exponential notation or exponential form is commonly found in algebraic terms, expressions, and equations. Exponents are used to shorten or condense repeated multiplication. For example, a term containing an exponent is shown below:

7⁴

In this term, 7 is the base and 4 is the exponent. The exponent or power, indicates the number of times that the factor, or base, is multiplied.

7⁴ = 7 • 7 • 7 • 7

When working with expressions containing only numbers, simply perform the indicated multiplication. Below are three examples:

15² = 15 · 15 = 225

3¹ = 3

2⁷ = 2 · 2 · 2 · 2 · 2 · 2 · 2 = 128

Notice in the second example that a number raised to the first power is just that number (3¹ = 3). When you see a number that does not have an exponent, it is because the 1 is assumed. When variables are included in the terms, we work according to the same principles.

Exponents in Algebra

Exponents included in algebraic terms and expressions can apply either to just one variable, an entire term, or an expression. When a term or expression is raised to a power, we must apply the exponent to the entire term or collection of terms.

A Variable Raised to a Power

If the term is simply one variable, the exponent is handled just as it was in the previous section. If we want to raise one number or variable to a power we put the exponent as a superscript to that one number or variable. An algebraic term containing an exponent is shown below.

In this term, x is the base, and 2 is the exponent. The exponent, or power, indicates the number of times that the factor, or base, occurs.

x²= x · x

In the example at the right, the term contains the variable to the fourth power. The numerical coefficient is not raised to the power. y is the base, and 4 is the exponent.

3y⁴ = 3 · y · y · y · y

As with numbers, a variable that does not appear to have a power is raised the first power.

z¹ = z

A Term Raised to a Power

Remember, a term is either a number or a product of a number and one or more variables. This includes any powers to which the variables are raised. In a term, either the entire term, or just one variable within the term, may be raised to a power. To indicate that the entire term is raised to a power, we must enclose the term in parentheses and place the exponent outside the parentheses. The general rule for applying an exponent to a whole term contained within parentheses is:

(x^a)^b = x^a•b

The result is much different than when the parentheses are omitted.

In the example at the right, the entire term is raised to the second power; this is commonly referred to as squared. Notice the parentheses around the entire term.

(5x)²= 5^{1 • 2}x^{1
• 2} =
5²x² = 5 · 5 · x · x = 25x²

Let's contrast the above example to the one at the right. In this example there are no parentheses grouping 5 and x. This means only the variable x is squared.

5x²= 5 · x · x

If we have a fraction inside parentheses raised to a power, the same rules apply. Each number and variable inside the parentheses must be raised to the power.

(2x³/y)² = 2^1•2x^3•2/ y^1•2 = 2²x⁶/y² = 4x⁶/y²

Multiplying Two Terms with the Same Variable

At times we must multiply together variables that are raised to different exponents. When multiplying together these types of terms, you add the exponents of each of the variables.

x^c • x^d = x^{c + d}

At first it may not be clear why this should be done. Look at an example of x^{2 ·} x⁴. If we expand both of these terms we have:

x²= x · x and x⁴= x · x · x · x

If we multiply these together we have:

x^{2 ·} x⁴ = x · x · x · x · x · x = x⁶ = x²⁺⁴

General Rules for Using Exponents with Variables and Terms

Any number or variable that appears to have no exponent is raised to the first power.
x¹ = x
If you raise any number or variable to the power of zero, the result is one.
5⁰ = 1, and z⁰ = 1
To apply an exponent to a term in exponential form, multiply the exponents.
(y^dx^a)^b = y^d•bx^a•b
An exponent outside the parentheses applies to all parts of a product or quotient inside the parentheses:
(xy)^a = (x • y)^a = x^a • y^a and (x/y)^a = x^a/y^a
When multiplying together two terms with the same variables, add the exponents.
x^c • x^d = x^{c + d}
A number or variable to a negative power means to move it to the denominator of a fraction and put it to the positive power.

Example

Multiply the following terms.

(4x³y)³
(3x²/5y²)⁴

Answers

First multiply the numerical coefficients, then use the exponent rules.

(4x³y)³ = 4³x^3•3y^1•3= 64x⁹y³
(3x²/5y²)⁴ = 3^1•4x^2•4/5^1•4y^2•4= 81x⁸/625y⁸

Exponents in Algebraic Expressions

If we have a collection of terms, called an expression, which is raised to an exponent, we again must apply the exponent to the entire expression. We place parentheses (or another grouping symbol) around the expression. The exponent is then placed outside the parentheses. Below is an example of an expression raised to a power:

(15x²+ xy)³= (15x²+ xy) (15x²+ xy) (15x²+ xy)

The next section will apply some of the basic rules of exponents used when multiplying and dividing algebraic terms.


[introduction]	[exponents]	[multiply/divide]	[add/subtract]	[grouping]

In this term, x is the base, and 2 is the exponent. The exponent, or power, indicates the number of times that the factor, or base, occurs.	x²= x · x
In the example at the right, the term contains the variable to the fourth power. The numerical coefficient is not raised to the power. y is the base, and 4 is the exponent.	3y⁴ = 3 · y · y · y · y

In the example at the right, the entire term is raised to the second power; this is commonly referred to as squared. Notice the parentheses around the entire term.	(5x)²= 5^{1 • 2}x^{1 • 2} = 5²x² = 5 · 5 · x · x = 25x²
Let's contrast the above example to the one at the right. In this example there are no parentheses grouping 5 and x. This means only the variable x is squared.	5x²= 5 · x · x