Basic Mathematics for College Students

The rules for natural-number exponents are summarized as follows.

Rules for Exponents

If and represent natural numbers and there are no divisions by zero, then

Exponent of 1 Product rule Power rule

Power of a product

(xy)nxnyn

x^1 x xmxnxmn (xm)nxmn

m n

8.6 Multiplication Rules for Exponents 693

Solution

a. Raise each factor of the product to the 4th power.

Evaluate:.

b. Raise each factor of the product to the 5th power.

For each power of a power, keep each base, and , and

multiply the exponents: 2  5 10 and 3  5 15.

x^10 y^15 x y

(x^2 y^3 )^5 (x^2 )^5 (y^3 )^5 x^2 y^3

 81 c^434  81

(3c)^4  34 c^43 c

EXAMPLE (^7) Simplify: (2a (^2) ) (^2) (4a (^3) ) 3
StrategyWe want to write an equivalent expression using one base and one
exponent. We will begin the process by using the power of a product rule for
exponents.
WHYWithin each set of parentheses is a product, and each product is raised to a
power.
Solution
(2a^2 )^2 (4a^3 )^3  22 (a^2 )^2  43 (a^3 )^3 Raise each factor of the product 2a^2 to the 2nd
power. Raise each factor of the product 4a^3 to
the 3rd power.
 4 a^4  64 a^9 Evaluate: 2^2 4 and 4^3 64. For each power
of a power, keep each base and multiply the
exponents: 2  2 4 and 3  3 9.
 (4  64)(a^4  a^9 ) Group the numerical factors. Group
the factors that have the same base.
 256 a^13 Do the multiplication: 4  64 256. Keep the
common base aand add the exponents: 4  9 13.

1. a.base: , exponent: 4 b.base: , exponent: 4 2. 3. a. b.
   c.(y1)^10 d. 4. a. b. 5. a. b. 6. a. b.


2. 432 y^18

s^8 t^7430 y^10 a^21 a^1716 t^4 c^18 d^24

y 3 y (xy)^5715 x^6

ANSWERS TO SELF CHECKS

Self Check 7

Simplify:

Now TryProblem 73

(4y^3 )^2 (3y^4 )^3

6

1

4

 4

256