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**5.1 Use Properties of Exponents**

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**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

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**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 =

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**Properties of Exponents Explained 3**

Power of a Product (ab) m a b m m = Example 4 (2 3) = 1296 = 2 3

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**Properties of Exponents Explained 4**

Negative Exponent 1 = -m (a) a m 1 1 Example -2 (7) = = 7 49 2

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**Properties of Exponents Explained 5**

Zero Exponent = 1 (a) Example (-89) = 1

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**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

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**Properties of Exponents Explained 7**

Quotient of Power m = Example 2 = =

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**Power of a product property**

EXAMPLE 1 Evaluate numerical expressions a. (– )2 = (– 4)2 (25)2 Power of a product property = Power of a power property = = 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.

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**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, Substitute values.

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**Write in scientific notation. Use multiplication properties.**

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 The number of locusts is about , or about 102,000,000,000. ANSWER

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GUIDED PRACTICE for Examples 1 and 2 Evaluate the expression. Tell which properties of exponents you used. (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

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GUIDED PRACTICE for Examples 1 and 2 – 4 4. 2 ANSWER ; quotient of power property

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**Product of powers property**

EXAMPLE 3 Simplify expressions a. b–4b6b7 = b– = 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 m4n –5 2n–5 = 8m4n –5 – (–5) Quotient of powers property = 8m4n0= 8m4 Zero exponent property

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**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

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**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

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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 (7y2z5)(y–4z–1) 7z4 y2 ; Product of powers property, Negative exponent property ANSWER

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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– x3y6 x3 y24 ; Quotient of powers property, Power of a Quotient property, Negative exponent property ANSWER

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