OBJECTIVE To understand that when numbers to a power are raised to a power, you multiply their powers
( y 2 ) 3 = y2 × y2 × y2 = y6
When you put an exponential expression to a power, you multiply the two powers. In general: (a p ) q = a p q
(103 )5 = 1015 (2 4 ) 3 = 212
( 3 ) 2 = 3 1
_ 2
1
(x−3)−2 = x6 (22x )−3 = 2− 6x
ACTION ACTIVITY
ACTION
Understanding powers of products and quotients
OBJECTIVE To manipulate more complicated expressions raised to a power
( y __
2 ) = 3
__ 2 ×
y
__ 2 ×
y
__ 2 =
y Y 21
5. Powers of products and quotients WORKED EXAMPLE
Y 20
4. The power of a power rule WORKED EXAMPLE
Powers of powers
Powers of products and quotients
(2y)4 = 2y × 2y × 2y × 2y = 2 × 2 × 2 × 2 × y × y × y × y = 24y4 y 3
________ 2 × 2 × 2 =
y × y × y
__ 2 3
For products and quotients raised to a power, put each factor to the power. In general: (ab) p = a pb p and
( a __
( )
__ b
( − 2 __
( xy __
( a 2 3
____ 4 y 2
2z ) = 3 x −2 2
) WARNING
Be careful never to apply this rule to a sum of terms raised to a power. (a + b) p ≠ a p + b p, except for p = 1. (a + b)2 ≠ a 2 + b 2 because (a + b)2 = a 2 + 2ab + b2.
