Free Algebra Tutorials!

Try the Free Math Solver or Scroll down to Tutorials!

 Depdendent Variable

 Number of equations to solve: 23456789
 Equ. #1:
 Equ. #2:

 Equ. #3:

 Equ. #4:

 Equ. #5:

 Equ. #6:

 Equ. #7:

 Equ. #8:

 Equ. #9:

 Solve for:

 Dependent Variable

 Number of inequalities to solve: 23456789
 Ineq. #1:
 Ineq. #2:

 Ineq. #3:

 Ineq. #4:

 Ineq. #5:

 Ineq. #6:

 Ineq. #7:

 Ineq. #8:

 Ineq. #9:

 Solve for:

 Please use this form if you would like to have this math solver on your website, free of charge. Name: Email: Your Website: Msg:

# Integral Exponents

In this section we will extend the definition of exponents to include all integers and to learn some rules for working with integral exponents.

## Positive and Negative Exponents

Positive integral exponents provide a convenient way to write repeated multiplication or very large numbers. For example,

2 Â· 2 Â· 2 = 23, y Â· y Â· y Â· y = y4, and 1,000,000,000 = 109.

We refer to 23 as â€œ2 cubed,â€ â€œ2 raised to the third power,â€ or â€œa power of 2.â€

Positive Integral Exponents

If a is a nonzero real number and n is a positive integer, then

In the exponential expression an, the base is a, and the exponent is n.

We use 2-3 to represent the reciprocal of 23. Because 23 = 8, we have . In general, a-n is defined as the reciprocal of an.

Negative Integral Exponents

If a is a nonzero real number and n is a positive integer, then

(If n is positive, -n is negative.)

To evaluate 2-3, you can first cube 2 to get 8 and then find the reciprocal to get , or you can first find the reciprocal of 2 (which is ) and then cube to get . So

The power and the reciprocal can be found in either order. If the exponent is -1, we simply find the reciprocal. For example,

Because 23 and 2-3 are reciprocals of each other, we have

These examples illustrate the following rules.

Rules for Negative Exponents

If a is a nonzero real number and n is a positive integer, then

Example 1

Negative exponents

Evaluate each expression.

a) 3-2

b) (-3)-2

c) -3-2

d)

e)

Solution

 Definition of negative exponent Definition of negative exponent Evaluate 3-2, then take the opposite. The reciprocal of .The cube of . The reciprocal of 5-3 is 53.

Caution

We evaluate -32 by squaring 3 first and then taking the opposite. So -32 = -9, whereas (-3)2 = 9. The same agreement also holds for negative exponents. That iswhy the answer to Example 1(c) is negative.