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# Dividing Monomials

Objective Learn the notion of dividing monomials, and the idea of zero and negative numbers as exponents.

It is a good idea that you experiment with computing quotients of powers of 2 in order to see if you can discover the formula for division of powers by yourself.

## Dividing Powers of the Same Number

To understand division of powers, look at the table that appears below. It shows the results when 64, which equals 2 6 , is divided by various powers of 2.

Do you see any pattern in these numbers? The result is found by subtracting the exponents.

Example 1

Divide x 4 by x 2 .

Solution

Expand x 4 into x Â· x Â· x Â· x and x 2 into x Â· x . Then use these expressions to write a fraction.

Now cancel two of the x ’ s from both numerator and denominator.

It is very important that you understand that “cancellation” just means recognizing that

You should understand that canceling is just shorthand for a process involving the properties of fractions. Any number raised to the power 1 is that number itself.

Quotient of Powers

If we divide a power of a number or variable by another (smaller) power of the same number or variable, the result is the original number raised to the power given by the difference of the two original powers.

Write the quotient and expand the numerator and the denominator into products of x’s.

This shows that the formula for the quotient of powers is valid.

Example 2

Simplify

Solution

This formula can be used to simplify any quotient of monomials.

Example 3

Simplify

Solution

## Negative Numbers and Zero in the Exponent

We have only defined exponentiation by positive integers, not for zero or for negative numbers. Remind students that when m > n ,

If we let m < n , then we get

On the other hand, , since any number divided by itself is 1. This leads to the following definition.

Zero Exponent

a 0 is defined to be equal to 1.

What would the formula suggest if m < n? For example, let m = 1 and n = 2.

On the other hand,

So, the formula suggests that a -1 should be defined to be . More generally, it suggests that a -n should be defined to be .

Negative Exponents

a -n is defined to be equal to .

At this point, you may be confused by the way the formula was used to generate a definition for negative powers. It was used only as a guide to suggest what a definition should be, but that once we make the definition, we can work with these exponents in exactly the same way that we did with positive exponents. With this definition, these exponents satisfy all the laws of exponents that you should already know. The definition of negative exponents can be used to simplify quotients of monomials.

Example 8

Simplify

Solution