Page 110 - MarceAlgebra Demystified

CHAPTER 5 Exponents and Roots 97

p p
ffiffiffi n n ffiffiffiffiffi n
Property 4 n a ¼ a ¼ a
Property 4 can be thought of as a root-power cancellation law.

Example

p p ffiffiffiffiffi p p ffiffiffiffiffiffiffi q ffiffiffiffiffiffiffiffiffiffiffi
3 ffiffiffiffiffi 3 3 ffiffiffi 2 3 3 3 3
27 ¼ 3 ¼ 3 ð 5Þ ¼ 5 8x ¼ ð2xÞ ¼ 2x

Practice

p ffiffiffiffiffiffiffiffiffiffi
1: 25x ¼
2
q ffiffiffiffiffiffiffi
3
2: 3 8y ¼
q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
3: ð4 xÞ ¼
q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3
4: 3 ½5ðx 1ފ ¼

Solutions

q ffiffiffiffiffiffiffiffiffiffiffi
p ffiffiffiffiffiffiffiffiffiffi
2
2
1: 25x ¼ ð5xÞ ¼ 5x
q ffiffiffiffiffiffiffi q ffiffiffiffiffiffiffiffiffiffi
3
3
2: 3 8y ¼ 3 ð2yÞ ¼ 2y
q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
3: ð4 xÞ ¼ 4 x
q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3
4: 3 ½5ðx 1ފ ¼ 5ðx 1Þ
These properties can be used to simplify roots in the same way canceling
is used to simplify fractions. For instance you normally would not leave
p ffiffiffiffiffi
25 without simplifying it as 5 any more than you would leave 12
p ffiffiffiffiffiffi 4
without reducing it to 3. In n a m if m is at least as large as n, then
p p p p
n
m
n
ffiffiffiffiffi
n ffiffiffiffiffiffi can be simplified using Property 1 ( ab ¼ n ffiffiffi ffiffiffi
a b) and Property 4
a
p ffiffiffiffiffi
n
n
( a ¼ a).