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 ⋅ 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
= 35 / 25
= (3 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 3) / (2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2)
= 243/32
Example 4 :
Evaluate :
34 / 3-3
Solution :
= 34 / 3-3
= 34 - (-3)
= 34 + 3
= 37
= 3 ⋅ 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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