Review of Using Exponents EXAMPLE 1 Review of Using Exponents Write 5 • 5 • 5 • 5 in exponential form, and find the value of the exponential expression. Since 5 appears as a factor 4 times, the base is 5 and the exponent is 4. Writing in exponential form, we have 5 4. 5 4 = 5 • 5 • 5 • 5 = 625

Evaluating Exponential Expressions EXAMPLE 2 Evaluating Exponential Expressions Evaluate each exponential expression. Name the base and the exponent. Base Exponent ( a ) 2 4 = 2 • 2 • 2 • 2 = 16 2 4 ( b ) – 2 4 = – ( 2 • 2 • 2 • 2 ) = – 16 2 4 ( c ) ( – 2 ) 4 = ( – 2 )( – 2 )( – 2 )( – 2 ) = 16 – 2 4

Understanding the Base CAUTION It is important to understand the difference between parts (b) and (c) of Example 2. In – 2 4 the lack of parentheses shows that the exponent 4 applies only to the base 2. In ( – 2 ) 4 the parentheses show that the exponent 4 applies to the base – 2. In summary, – a m and ( – a ) m mean different things. The exponent applies only to what is immediately to the left of it. Expression Base Exponent Example – a n a n – 5 2 = – ( 5 • 5 ) = – 25 ( – a ) n – a n ( – 5 ) 2 = ( – 5 ) ( – 5 ) = 25

Product Rule for Exponents If m and n are positive integers, then a m • a n = a m + n (Keep the same base and add the exponents.) Example: 3 4 • 3 2 = 3 4 + 2 = 3 6

Common Error Using the Product Rule CAUTION Avoid the common error of multiplying the bases when using the product rule. Keep the same base and add the exponents. 3 4 • 3 2 ≠ 9 6 3 4 • 3 2 = 3 6

Using the Product Rule EXAMPLE 3 Using the Product Rule Use the product rule for exponents to find each product, if possible. ( a ) 6 2 • 6 7 = 6 2 + 7 = 6 9 by the product rule. ( b ) ( – 7 ) 1 ( – 7 ) 5 ( b ) ( – 7 ) 1 ( – 7 ) 5 = ( – 7 ) 1 + 5 = ( – 7 ) 6 by the product rule. ( c ) 4 7 • 3 2 The product rule doesn’t apply. The bases are different. ( d ) x 9 • x 5 = x 9 + 5 = x 14 by the product rule.

Using the Product Rule EXAMPLE 3 Using the Product Rule Use the product rule for exponents to find each product, if possible. ( e ) 8 2 + 8 3 The product rule doesn’t apply because this is a sum. ( f ) ( 5 m n 4 ) ( – 8 m 6 n 11 ) = ( 5 • – 8 ) • ( m m 6 ) • ( n 4 n 11 ) using the commutative and associative properties. = – 40 m 7 n 15 by the product rule.

Product Rule and Bases CAUTION The bases must be the same before we can apply the product rule for exponents.

Understanding Differences in Exponential Expressions CAUTION Be sure you understand the difference between adding and multiplying exponential expressions. Here is a comparison. Adding expressions 3 x 4 + 2 x 4 = 5 x 4 Multiplying expressions ( 3 x 4 ) ( 2 x 5 ) = 6 x 9

Power of a Power Rule for Exponents Power of a Power for Exponents If m and n are positive integers, then ( a m ) n = a m n (Raise a power to a power by multiplying exponents.) Example: ( 3 5 ) 2 = 3 5 • 2 = 3 10

Using Power of a Power Rule EXAMPLE 4 Using Power of a Power Rule Use power of a power rule to simplify each expression. Write answers in exponential form. ( a ) ( 3 2 ) 7 = 3 2 • 7 = 3 14 ( b ) ( 6 5 ) 9 = 6 5 • 9 = 6 45 ( c ) ( w 4 ) 2 = w 4 • 2 = w 8

Power of a Product Rule for Exponents If m is a positive integer, then ( a b ) m = a m b m (Raise a product to a power by raising each factor to the power.) Example: ( 5a ) 8 = 5 8 a 8

Using Power of a Product Rule EXAMPLE 5 Using Power of a Product Rule Use power of a product rule to simplify each expression. ( a ) ( 4n ) 7 = 4 7 n 7 ( b ) 2 ( x 9 y 4 ) 5 = 2 ( x 45 y 20 ) = 2 x 45 y 20 ( c ) 3 ( 2 a 3 b c 4 ) 2 = 3 ( 2 2 a 6 b 2 c 8 ) = 3 ( 4 a 6 b 2 c 8 ) = 12 a 6 b 2 c 8

The Power of a Product Rule CAUTION Power of a product rule does not apply to a sum. ( x + 3 ) 2 ≠ x 2 + 3 2  Error You will learn how to work with ( x + 3 ) 2 in more advanced mathematics courses.

Power of a Quotient Rule for Exponents If m is a positive integer, then = (Raise a quotient to a power by raising both the numerator and the denominator to the power. The denominator cannot be 0.) Example: = a b m a m b m 3 4 2 3 2 4 2

Using Power of a Quotient Rule EXAMPLE 6 Using Power of a Quotient Rule Simplify each expression. ( a ) 5 8 3 = 5 3 8 3 = 125 512 ( b ) 3a 9 7 b c 3 2 ( 3a 9 ) 2 ( 7 b 1 c 3 ) 2 = 3 2 a 18 7 2 b 2 c 6 = 9 a 18 49 b 2 c 6 =

If a is any nonzero number, then, a 0 = 1. Example: 25 0 = 1 Zero Exponent Zero Exponent If a is any nonzero number, then, a 0 = 1. Example: 25 0 = 1 17

EXAMPLE 1 Using Zero Exponents Evaluate each exponential expression. ( a ) 31 0 = 1 ( b ) ( – 7 ) 0 = 1 ( c ) – 7 0 = – ( 1 ) = – 1 ( d ) g 0 = 1, if g ≠ 0 ( e ) 5n 0 = 5 ( 1 ) = 5, if n ≠ 0 ( f ) ( 9v ) 0 = 1, if v ≠ 0 18

Notice the difference between parts (b) and (c) from Example 1. Zero Exponents CAUTION Notice the difference between parts (b) and (c) from Example 1. In Example 1 (b) the base is – 7 and in Example 1 (c) the base is 7. ( b ) ( – 7 ) 0 = 1 The base is – 7. ( c ) – 7 0 = – ( 1 ) = – 1 The base is 7. 19

If a is any nonzero real number and n is any integer, then Negative Exponents Negative Exponents If a is any nonzero real number and n is any integer, then Example: 1 a n a – n = 1 7 2 7 – 2 = 20

Using Negative Exponents EXAMPLE 2 Using Negative Exponents Simplify by writing each expression with positive exponents. Then evaluate the expression. 1 8 2 = 1 64 = ( a ) 8 –2 1 5 1 = 1 5 = ( b ) 5 –1 1 n 8 = ( c ) n –8 when n ≠ 0 21

Using Negative Exponents EXAMPLE 2 Using Negative Exponents Simplify by writing each expression with positive exponents. Then evaluate the expression. 1 3 1 = 2 1 + ( d ) 3 –1 + 2 –1 Apply the exponents first. 1 3 = 2 + 2 6 = 3 + Get a common denominator. 5 6 = Add. 22

Negative Exponent Example CAUTION A negative exponent does not indicate a negative number; negative exponents lead to reciprocals. Expression Example 1 7 2 7 – 2 = 49 a – n Not negative 23

Quotient Rule for Exponents If a is any nonzero real number and m and n are any integers, then (Keep the base and subtract the exponents.) Example: a m a n a m – n = 3 8 3 2 3 8 – 2 = 3 6 = 24

A common error is to write . CAUTION 3 8 3 2 1 8 – 2 = 1 6 A common error is to write . When using the rule, the quotient should have the same base. The base here is 3. If you’re not sure, use the definition of an exponent to write out the factors. 3 8 3 2 3 8 – 2 = 3 6 1 1 3 8 3 2 = 3 • 3 • 3 • 3 • 3 • 3 • 3 • 3 3 • 3 3 6 = 1 1 1 25

Using the Quotient Rule for Exponents EXAMPLE 3 Using the Quotient Rule for Exponents Simplify using the quotient rule for exponents. Write answers with positive exponents. 4 7 4 2 ( a ) 4 7 – 2 = 4 5 2 3 2 9 = 1 2 6 ( b ) 2 3 – 9 = 2 –6 9 –3 9 –6 ( c ) 9 –3 – (–6) = 9 3 26

Using the Quotient Rule for Exponents EXAMPLE 3 Using the Quotient Rule for Exponents Simplify using the quotient rule for exponents. Write answers with positive exponents. x 8 x –2 ( d ) x 8 – (–2) = x 10 when x ≠ 0 n –7 n –4 = 1 n 3 ( e ) n –7 – (–4) = n –3 when n ≠ 0 r –1 r 5 1 r 6 = ( f ) r –1 – 5 = r –6 when r ≠ 0 27

Using the Product Rule with Negative Exponents EXAMPLE 4 Using the Product Rule with Negative Exponents Simplify each expression. Assume all variables represent nonzero real numbers. Write answers with positive exponents. ( a ) 5 8 (5 –2 ) 5 8 + (–2) = 5 6 = = 1 6 7 ( b ) (6 –1 )(6 –6 ) 6 (–1) + (–6) = 6 –7 = = 1 g 2 ( c ) g –4 • g 7 • g –5 g (–4) + 7 + (–5) = g –2 = 28

Definitions and Rules for Exponents If m and n are positive integers, then Product Rule a m • a n = a m + n 3 4 • 3 2 = 3 4 + 2 = 3 6 Power of a Power Rule ( a m ) n = a m n ( 3 5 ) 2 = 3 5 • 2 = 3 10 Power of a Product Rule ( a b ) m = a m b m ( 5a ) 8 = 5 8 a 8 Power of a Quotient Rule ( b ≠ 0 ) Examples a m b m a b m = 3 2 4 2 3 4 2 =

Definitions and Rules for Exponents If m and n are positive integers and when a ≠ 0, then Zero Exponent a 0 = 1 (–5) 0 = 1 Negative Exponent Quotient Exponent Examples 1 a n a – n = 1 4 2 4 – 2 = a m a n a m – n = 2 3 2 8 2 3 – 8 = 2 –5 = 1 2 5 = 30

Simplifying Expressions vs Evaluating Expressions NOTE Make sure you understand the difference between simplifying expressions and evaluating them. Example: Simplifying Evaluating 2 3 2 8 1 2 5 = 1 32 = 2 3 – 8 = 2 –5 = 31