EVALUATING POWERS

The following properties of exponents can be used to evaluate expressions with powers. 

xm ⋅ xn  =  xm+n

xm ÷ xn  =  xm-n

(xm)n  =  xmn

(xy)m  =  xm ⋅ ym

(x / y)m  =  xm / ym

x-m  =  1 / xm

xm/n  =  y -----> x  =  yn/m

(x / y)-m  =  (y / x)m

Example 1 :

Evaluate : 

43  53

Solution :

=  43  53

=  ⋅ 4 ⋅ 4 ⋅ 5 ⋅ 5 ⋅ 5

=  64  125

=  8000

Example 2 :

Evaluate :

109 ÷ 106

Solution :

=  109 ÷ 106

=   109 - 6

=   103

=  10  10  10

=  1000

Example 3 :

Evaluate :

(3/2)5

Solution :

=  (3/2)5

=  3/ 25 

  =  (3  3  3  3  3) / (2  2  2  2  2)

  =  243/32

Example 4 :

Evaluate : 

3/ 3-3

Solution :

=  3/ 3-3

=  34 - (-3)

=  34 + 3

=  37

=   3  3  3  3 ⋅ 3 ⋅ 3

=  2187

Example 5 :

Evaluate the following expression when p = 2.

(4p)3  (2p)2  p

Solution :

=  (4p)3  (2p)2  p

=  (43p3)  (22p2)  p

=  (4 ⋅ 4 ⋅ 4 ⋅ p3)  (2 ⋅ 2 ⋅ p2)  p

=  (64p3)  (4p2)  p

=  (64 ⋅ 4) ⋅ (p3  p2  p)

=  256 p3 + 2 + 1

=  256 ⋅ p6

Substitute 2 for p. 

=  256 ⋅ 26

=  256 ⋅ (2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2)

=  256 ⋅ 64

=  16384

Example 6 :

Evaluate : 

[(-2/3)-2]-2

Solution :

=  [(-2/3)-2]-2

=  (-2/3)(-2)(-2)

=  (-2/3)4

=  (-2)4 / 34

=  (-2   -2   -2  -2) / (3  3  3  3)

=  16 / 81

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