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# Dividing Monomials Using the Quotient Rule

You problably know the definition of division to divide signed numbers. Because the definition of division applies to any division, we restate it here.

Division of Real Numbers

If a, b and c are any numbers with b ≠ 0, then

a Ã· b = c provided that c Â· b = a.

If a Ã· b = c, we call a the dividend, b the divisor, and c (or a Ã· b) the quotient.

You can find the quotient of two monomials by writing the quotient as a fraction and then reducing the fraction. For example,

You can be sure that x3 is correct by checking that x3 Â· x2 = x5. You can also divide x2 by x5, but the result is not a monomial:

Note that the exponent 3 can be obtained in either case by subtracting 5 and 2. These examples illustrate the quotient rule for exponents.

Quotient Rule

Suppose a ≠ 0, and m and n are positive integers.

If m ≥ n, then

If n > m, then

Note that if you use the quotient rule to subtract the exponents in x4 Ã· x4, you get the exporession x4 - 4, or x0, which has not been defined yet. Because we must have x4 Ã· x4 = 1 if x ≠ 0, we define the zero power of a nonzero real number to be 1. We do not define the expression 00.

Zero Exponent

For any nonzero real number a, a0 = 1.

Example 1

Using the definition of zero exponent

Simplify each expression. Assume that all variables are nonzero real numbers.

a) 50

b) (3xy)0

c) a0 + b0

Solution

a) 50 = 1

b) (3xy)0 = 1

c) a0 + b0 = 1 + 1 = 2

With the definition of zero exponent the quotient rule is valid for all positive integers as stated.

Example 2

Using the quotient rule in dividing monomials

Find each quotient.

Solution

Use the definition of division to check that y4 Â· y5 = y9.

Use the definition of division to check that

Use the definition of division to check that

Use the definition of division to check that x6 Â· x2y2 = x8y2.