5.1 Use Properties of Exponents

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 ·

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 =

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

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

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

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

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

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

Substitute values. 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 square kilometers. About how many locusts are in such a swarm? SOLUTION = 85,000,000 1200 Substitute values.

Write in scientific notation. Use multiplication properties. EXAMPLE 2 Use scientific notation in real life = (8.5 107)(1.2 103) Write in scientific notation. = (8.5 1.2)(107 103) Use multiplication properties. = 10.2 1010 Product of powers property = 1.02 101 1010 Write 10.2 in scientific notation. = 1.02 1011 Product of powers property The number of locusts is about 1.02 1011, or about 102,000,000,000. ANSWER

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

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

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

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

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

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

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