5.1 Use Properties of Exponents

1. 5.1 Use Properties of Exponents

2. Properties of Exponents Explained 1 Product of Powers a a a · n = m+ n m Example 5 3 5 5 3 + (-1) 5 -1 = = 2 = 25 ·

3. Properties of Exponents Explained 2 Power of a Power · (a ) m n = a m n Example · 2 (3 ) 3 3 2 3 6 = 729 = 3 =

4. Properties of Exponents Explained 3 Power of a Product (ab) m a b m m = Example 4 (2 3) 4 4 = 1296 · = 2 3 ·

5. Properties of Exponents Explained 4 Negative Exponent 1 = -m (a) a m 1 1 Example -2 (7) = = 7 49 2

6. Properties of Exponents Explained 5 Zero Exponent = 0 1 (a) Example 0 (-89) = 1

7. Properties of Exponents Explained 6 Quotient of Powers a m m - n = a a n Example -3 6 -3 – (-6) 3 = 216 = = 6 6 -6 6

8. Properties of Exponents Explained 7 Quotient of Power m = Example 2 = =

9. a. (–4 25)2 = 16 252 = 16,384 = 16 210 = (– 4)2(25)2 –1 115 118 = b. 118 115 = 1331 = 113 EXAMPLE 1 Evaluate numerical expressions Power of a product property Power of a power property Simplify and evaluate power. Negative exponent property = 118–5 Quotient of powers property Simplify and evaluate power.

10. = 85,000,000 1200 EXAMPLE 2 Use scientific notation in real life Locusts A swarm of locusts may contain as many as 85 million locusts per square kilometer and cover an area of 1200 squarekilometers. About how many locusts are in such a swarm? SOLUTION Substitute values.

11. = (8.5 107)(1.2 103) = (8.5 1.2)(107 103) = 10.21010 = 1.02 1011010 = 1.02 1011 ANSWER The number of locusts is about1.02 1011, or about102,000,000,000. EXAMPLE 2 Use scientific notation in real life Write in scientific notation. Use multiplication properties. Product of powers property Write 10.2 in scientific notation. Product of powers property

12. 2 3 3. 9 for Examples 1 and 2 GUIDED PRACTICE Evaluate the expression. Tell which properties of exponents you used. 1. (42 )3 4096 ; Power of a power property ANSWER 2.(–8)(–8)3 4096 ; Product of a powers property ANSWER 8 ANSWER ; Power of a quotient property 729

13. 6 10 – 4 9 107 4. for Examples 1 and 2 GUIDED PRACTICE 2 3 1011 ANSWER ; quotient of power property

14. = b–4 +6+7 = b9 r–2–3 ( r –2 )–3 ( s3 )–3 = b. s3 r 6 = s–9 c. 16m4n –5 2n–5 EXAMPLE 3 Simplify expressions Product of powers property a. b–4b6b7 Power of a quotient property Power of a power property = r6s9 Negative exponent property Quotient of powers property = 8m4n –5 – (–5) = 8m4n0= 8m4 Zero exponent property

15. (x–3)2(y3)2 (x–3y3)2 = x5y6 x5y6 x –6y6 = x5y6 EXAMPLE 4 Standardized Test Practice SOLUTION Power of a product property Power of a power property

16. = x–111 1 = x11 The correct answer is B. ANSWER EXAMPLE 4 Standardized Test Practice = x –6– 5y6– 6 Quotient of powers property = x–11y0 Simplify exponents. Zero exponent property Negative exponent property

17. 7z4 y2 for Examples 3, 4, and 5 GUIDED PRACTICE Simplify the expression. Tell which properties of exponents you used. 5. x–6x5 x3 x2 ;Product of powers property ANSWER 6. (7y2z5)(y–4z–1) ; Product of powersproperty, Negative exponent property ANSWER

18. x4y–2 3 s 3 2 7. 8. t–4 x3y6 s6t8 x3 y24 for Examples 3, 4, and 5 GUIDED PRACTICE ANSWER ; Power of a power property, Negative exponent property ; Quotient of powers property, Power of a Quotient property, Negative exponent property ANSWER