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Columbus State Community College Chapter 8 Section 2 Integer Exponents and the Quotient Rule

Integer Exponents and the Quotient Rule • Use 0 as an exponent. • Use negative numbers as exponents. • Use the quotient rule for exponents. • Use the product rule with negative exponents.

Zero Exponent Zero Exponent If a is any nonzero number, then, a0 = 1. Example: 250= 1

Using Zero Exponents EXAMPLE 1 Using Zero Exponents Evaluate each exponential expression. ( a ) 310 = 1 ( b ) ( –7 )0 = 1 ( c ) –70 = – ( 1 ) = – 1 ( d ) g0 = 1,if g ≠ 0 ( e ) 5n0 = 5 ( 1 ) = 5, if n≠ 0 ( f ) ( 9v )0 = 1, if v≠ 0

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 ) –70 = – ( 1 ) = – 1 The base is 7.

1 7 – 2 = 72 1 a– n = an Negative Exponents Negative Exponents If a is any nonzero real number and n is any integer, then Example:

1 1 1 1 1 = = = = = ( c ) ( b ) ( a ) 5 –1 8 –2 n–8 51 82 5 n8 64 Using Negative Exponents EXAMPLE 2 Using Negative Exponents Simplify by writing each expression with positive exponents. Then evaluate the expression. when n ≠ 0

1 3 2 5 1 1 1 + = = = + + = ( d ) 3 –1 + 2 –1 2 6 6 6 21 3 31 Using Negative Exponents EXAMPLE 2 Using Negative Exponents Simplify by writing each expression with positive exponents. Then evaluate the expression. Apply the exponents first. Get a common denominator. Add.

1 1 7 – 2 = = 72 49 Negative Exponent CAUTION A negative exponent does not indicate a negative number; negative exponents lead to reciprocals. Expression Example a– n Not negative

am = am – n an 38 = 38 – 2 = 36 32 Quotient Rule for Exponents 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:

38 38 38 3 • 3 • 3 • 3 • 3 • 3 • 3 • 3 36 = = 18 – 2 38 – 2 = = 16 36 = = = 36 32 32 32 3 • 3 1 Common Error CAUTION 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. 1 1 1 1

1 23 9–3 47 = = 23 – 9 47 – 2 = = 45 2–6 26 29 9–6 42 = = 9–3 – (–6) = 93 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. ( a ) ( b ) ( c )

n–7 r–1 1 1 x8 = = x8 – (–2) n–7 – (–4) = = x10 n–3 r6 x–2 n3 r5 n–4 = = r–1 – 5 = = r–6 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. ( d ) when x≠ 0 ( e ) when n≠ 0 when r≠ 0 ( f )

1 1 58 (5–2) ( a ) = 58 + (–2) = 56 g2 67 = ( b ) = 6(–1) + (–6) = 6–7 (6–1 )(6–6 ) ( c ) = g–2 g–4• g7• g–5 = g(–4) + 7 + (–5) = 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.

2 3 32 a a m m 4 42 = = b b m Definitions and Rules for Exponents Definitions and Rules for Exponents If m and n are positive integers, then Product Ruleam • an = am + n34 • 32 = 34 + 2 = 36 Power Rule (a)( am )n = am n( 35 )2 = 35 • 2 = 310 Power Rule (b)( a b )m = am bm( 5a )8 = 58 a8 Power Rule (c)(b≠ 0 ) Examples

1 a– n = an 1 4 – 2 = 42 am = am – n 23 an = 23 – 8 = 2–5 28 1 = 25 Definitions and Rules for Exponents Definitions and Rules for Exponents If m and n are positive integers and when a ≠ 0, then Zero Exponenta0 = 1 (–5)0= 1 Negative Exponent Quotient Exponent Examples

23 1 1 = 23 – 8 = 2–5 = = 28 25 32 Simplifying Expressions vs Evaluating Expressions NOTE Make sure you understand the difference between simplifying expressions and evaluating them. Example: Simplifying Evaluating

Integer Exponents and the Quotient Rule Chapter 8 Section 2 – Completed Written by John T. Wallace

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