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# Multiplication Property of Exponents

Exponents have several properties. We will use these properties to simplify expressions. In the properties that follow, each variable represents a real number.

## Multiplication Property

Property â€” Multiplication Property of Exponents

English To multiply two exponential expressions with the same base, add their exponents. The base stays the same.

Algebra xm Â· xn = xm + n (Here, m and n are positive integers.)

Example 54 Â· 52 = 54 + 2 = 56

Example 1

a. Use the Multiplication Property of Exponents to simplify 23 Â· 24.

Solution

a. The operation is multiplication and the bases are the same. Therefore, add the exponents and use 2 as the base.

23 Â· 24 = 23 + 4 = 27

b. Rewrite the product to show the factors. Then simplify.

Note:

Remember to add the exponents, but leave the bases alone.

That is, 23 Â· 24 = 23 + 4 = 27 , not 47.

Note the difference between 23 Â· 24 and 23 + 24.

23 Â· 24 = 23 + 4 = 27 = 128

23 + 24 = 8 + 16 = 24

Caution â€” Negative Bases

A negative sign is part of the base only when the negative sign is inside the parentheses that enclose the base.

For example, consider the following cases:

In (-3)2, the base is -3.

(-3)2 = (-3) Â· (-3) = +9

In -32, the base is 3.

You can think of -32 as the â€œoppositeâ€ of 32.

-32 = -(3 Â· 3) = -9

Example 2

If possible, use the Multiplication Property of Exponents to simplify each expression:

a. ( 2)2 Â· ( 2)4

b. 22 Â· 24

c. 22 Â· ( 2)4

 a. In (-2)2 Â· (-2)4, the base is -2. (-2)2 Â· (-2)4 = (-2)2 + 4 = (-2)6 = 64 b. In -22 Â· 24, the base is 2. We may think of -22 Â· 24 as the opposite of 22 Â· 24. -22 Â· 24 = -(22) Â· (24) = -(22 + 4) = -(26) = -64

c. In -22 Â· (-2)4, the base of the first factor, -22, is 2.

The base of the second factor, (-2)4, is -2.

The bases are not the same, so we cannot use the Multiplication Property of Exponents.

However, we can still evaluate the expression. -22 Â· (-2)4 = -4 Â· 16 = -64