1. Dividing Monomials (7-2) Objective: Find the quotient of two monomials. Simplify expressions containing negative and zero exponents.

2. Quotients of Monomials We can use the principles for reducing fractions to find quotients of monomials. In the following examples, look for a pattern in the exponents.

3. Quotient of Powers To divide two powers with the same base, subtract the exponents. For any nonzero number a, and any integers m and p,

4. Example 1 Simplify. Assume that no denominator equals zero. = x 7-6 ∙ y 12-3 = xy 9

5. Check Your Progress Choose the best answer for the following. –Simplify. Assume that a and b are not equal to zero. A.a 4 b 11 B.. C.. D.a 2 b 7 = a 3-1 ∙ b 9-2

6. Powers of Quotients We can use the Product of Powers Rule to find the powers of quotients for monomials. In the following examples, look for a pattern in the exponents.

7. Power of a Quotient To find the power of a quotient, find the power of the numerator and the power of the denominator. For any real numbers a and b ≠ 0, and any integer m,

8. Example 2 Simplify

9. Check Your Progress Choose the best answer for the following. –Simplify. Assume that p and q are not equal to zero. A.. B.. C.. D..

10. Zero Exponent Property A zero exponent is any nonzero number raised to the zero power. Any nonzero number raised to the zero power is equal to 1. For any nonzero number a, a 0 = 1. 15 0 =1 1 1

11. Example 3 Simplify each expression. Assume that no denominator equals zero. = 1

12. Check Your Progress Choose the best answer for the following. A.Simplify. Assume that z is not equal to zero. A. 3 / 5 B.1 C.0 D.-1

13. Check Your Progress Choose the best answer for the following. B.Simplify. Assume that x and k are not equal to zero. A.xk 2 B.. C.(xk) 2 D.k 2

14. Negative Exponents Any nonzero real number raised to a negative power is a negative exponent. Look for a pattern in the following example.

15. Negative Exponent Property For any nonzero number a and any integer n, a -n is the reciprocal of a n. Also, the reciprocal of a -n is a n. For any nonzero number a and any integer n, a -n = and = a n.

16. Negative Exponent Property An expression is considered simplified when it contains only positive exponents, each base appears exactly once, there are no powers of powers, and all fractions are in simplest form.

17. Example 4 Simplify each expression. Assume that no denominator equals zero.

18. Example 4 Simplify each expression. Assume that no denominator equals zero.

19. Check Your Progress Choose the best answer for the following. A.Simplify. Assume that no denominator is equal to zero. A.. B.. C.. D..

20. Check Your Progress Choose the best answer for the following. B.Simplify. Assume that no denominator is equal to zero. A.. B.. C.. D..

21. Apply Properties of Exponents Order of magnitude is used to compare measures and to estimate and perform rough calculations. The order of magnitude of a quantity is the number rounded to the nearest power of 10. For example, the power of 10 closest to 95,000,000,000 is 10 11, or 100,000,000,000. So the order of magnitude of 95,000,000,000 is 10 11.

22. Example 5 Darin has $123,456 in his savings account. Tabo has $156 in his savings account. Determine the order of magnitude of Darin’s account and Tabo’s account. How many orders of magnitude as great is Darin’s account as Tabo’s account? –Darin: 123,456  100,000 = 10 5 –Tabo: 156  100 = 10 2 –Darin’s account is 1000 times greater or greater by a magnitude of 10 3.

23. Check Your Progress Choose the best answer for the following. –A circle has a radius of 210 centimeters. How many orders of magnitude as great is the area of the circle as the circumference of the circle? A.10 1 B.10 2 C.10 3 D.10 4 A =  r 2 =  (210) 2 = 138,544  100,000 = 10 5 C = 2  r = 2  (210) = 1319  1000 = 10 3