1. Math 1B
Exponent Rules

2. Obj: To simplify expressions with zero and negative exponents.
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3. Vocabulary PROPERTY: Zero as an Exponent: For every nonzero number a,
Negative Exponents: For every nonzero number a and integer n, a-n = 1/an

4. Demonstrate why #^0 is 1 20 30

5. Zero and Negative Exponents
Simplify. =
Use the definition of negative exponent. 1 32 a.
3–2
Simplify. 1 9 = b.
(–22.4)0
Use the definition of zero as an exponent.
= 1

6. Zero and Negative Exponents
Simplify 1
x –3 a.
Rewrite using a division symbol.
= 1  x –3
3ab –2 1 b2
Use the definition of negative exponent.
= 3a b.
= 1  1
x 3
Use the definition of negative exponent.
Simplify. 3a
b 2 =
= 1 • x 3
Multiply by the reciprocal of , which is x 3. 1 x3
= x 3
Identity Property of Multiplication
8-1

7. Zero and Negative Exponents
Evaluate 4x 2y –3 for x = 3 and y = –2.
Method 1: Write with positive exponents first.
4x 2y –3 =
Use the definition of negative exponent.
4x 2
y 3
Substitute 3 for x and –2 for y.
4(3)2
(–2)3 = 36 –8 –4 1 2 =
Simplify.
8-1

8. Zero and Negative Exponents
(continued)
Method 2: Substitute first.
4x 2y –3 = 4(3)2(–2)–3
Substitute 3 for x and –2 for y.
4(3)2
(–2)3 =
Use the definition of negative exponent. 36 –8 –4 1 2 =
Simplify.
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9. Vocabulary
Multiplying Powers with the Same Base: For every nonzero number a and integers m and n,
am * an = am+n

10. Multiplication Properties of Exponents
Rewrite each expression using each base only once.
Add exponents of powers with the
same base.
73 + 2 = a.
73 • 72
= 75
Simplify the sum of the exponents.
Think of – 2 as (–2) to add the exponents.
– 2 = b.
44 • 41 • 4–2
= 43
Simplify the sum of the exponents.
Add exponents of powers with the
same base.
68 + (–8) = c.
68 • 6–8
= 60
Simplify the sum of the exponents.
Use the definition of zero as an exponent.
= 1
8-3

11. Multiplication Properties of Exponents
Simplify each expression. a.
p2 • p • p5
Add exponents of powers with the same base. p =
= p 8
Simplify.
4x6 • 5x–4 b.
Commutative Property of Multiplication
(4 • 5)(x 6 • x –4) =
Add exponents of powers with the same base.
= 20(x 6+(–4))
Simplify.
= 20x 2
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12. Multiplication Properties of Exponents
Simplify each expression. a.
a 2 • b –4 • a 5
Commutative Property of Multiplication
a 2 • a 5 • b –4 =
= a • b –4
Add exponents of powers with the same base.
Simplify.
a 7
b 4 =
Commutative and Associative Properties of Multiplication
(2 • 3 • 4)(p 3)(q • q 4) = b.
2q • 3p3 • 4q4
= 24(p 3)(q 1 • q 4)
Multiply the coefficients. Write q as q 1.
= 24(p 3)(q 1 + 4)
Add exponents of powers with the same base.
= 24p 3q 5
Simplify.
8-3

13. Division Properties of Exponents
Simplify each expression. x4 x9 =
Subtract exponents when dividing powers with the same base.
x4 – 9 a.
Simplify the exponents.
= x–5
Rewrite using positive exponents. 1 x5 =
p3 j –4
p–3 j 6 =
Subtract exponents when dividing powers with the same base.
p3 – (–3)j –4 – 6 b.
= p6 j –10
Simplify.
Rewrite using positive exponents. p6
j10 =
8-5

14. Division Properties of Exponents2 3 –3
a. Simplify . 2 3 –3 =
Rewrite using the reciprocal of . 33 23 =
Raise the numerator and the denominator to the third power.
Simplify. 27 8 3 or =
8-5

15. Division Properties of Exponents4b c –
b. Simplify 4b c – –2 2 =
Rewrite using the reciprocal of .
Write the fraction with a negative numerator. c 4b – 2 =
Raise the numerator and denominator to the second power.
(–c)2
(4b)2 =
Simplify. c2
16b2 =