Basic Mathematics for College Students

Chapter 8 Summary and Review 705

SECTION 8.6 Multiplication Rules for Exponents

An exponentindicates repeated multiplication. It

tells how many times the baseis to be used as a

factor.

Exponent factors of

Base



xnxxxpx

⎫⎪⎪⎬⎪⎪⎭

 n x

DEFINITIONS AND CONCEPTS EXAMPLES

Identify the base and the exponent in each expression.

2 is the base and 6 is the exponent.

Because of the parentheses, is the

base and 3 is the exponent.

The base is and 4 is the exponent.

81  8 The base is 8 and 1 is the exponent.

5 t^4  5 tttt t

(xy)^3 (xy)(xy)(xy) xy

26  2  2  2  2  2  2

Rules for Exponents:If and represent

integers,

Product rule:

Power rule:

Power of a product rule:(xy)mxmym

(xm)nxm^ ^ nxmn

xmxnxmn

m n Simplify each expression:

Keep the common base, 5, and add the exponents.

Keep the base, 6, and multiply the exponents.

(2p)^5  25 p^5  32 p^5 Raise each factor of the product 2pto the 5th power.

(6^3 )^7  63 ^7  621

5257  52 ^7  59

To simplify some expressions, we must apply

two (or more) rules for exponents.

Simplify: (c^2 c^5 )^4 (c^7 )^4 Within the parentheses, keep the common base, c,

and add the exponents: 2  5 7.

c^28 Keep the base, c, and multiply the exponents:

7  4 28.

Simplify: (t^2 )^4 (t^3 )^3 t^8 t^9 For each power of t raised to a power, keep the

base and multiply the exponents: 2  4 8 and

3  3 9.

t^17 Keep the common base, t, and add the exponents:

8  9 17.

75. Identify the base and the exponent in each
    expression.
    a. b.
    c. d.


76. Write each expression in an equivalent form using
    an exponent.
    a. b.
    c. d.


77. Simplify, if possible.
    a. b.
    c. xx^2 d. xx^2

x^2 x^2 x^2 x^2

aabbbb (pq)(pq)(pq)

mmmmm  3 xxxx

3 r^4 (y7)^3

n^12 (2x)^6

78. Explain each error.
    a.
    b.

Simplify each expression.

79. 
    
    
    
    80. 
        
    
    
    
    


80. 
    
    
    
    82. 
        
    
    
    
    


81. 
    
    
    
    84. 
        
    
    
    
    


82. 
    
    
    
    86. 
        
    
    
    
    


83. 
    
    
    
    88. 
        
    
    
    
    


84. 
    
    
    
    90. 
        
    
    
    
    


85. 
    
    
    
    92. 
        
    
    
    
    


86. (4m^3 )^3 (2m^2 )^2 94. (3t^4 )^3 (2t^5 )^2

(3a^4 )^2 (2a^3 )^3 x^100 x^100

(2x^2 x^3 )^3 (m^2 m^3 )^2 (n^2 n^4 )^3

[(9)^3 ]^5 (a^5 )^3 (a^2 )^4

( 16 s^3 )^2 s^4 (2.1x^2 y)^2

(6^3 )^12 b^3 b^4 b^5

(y^7 )^3 (3x)^4

74  78 mmnn^2

(3^2 )^4  36

32  34  96

REVIEW EXERCISES