Page 109 - MarceAlgebra Demystified

96 CHAPTER 5 Exponents and Roots

p ffiffiffi n
general, n a ¼ b if b ¼ a. There is no problem with odd roots being
negative numbers:
p ffiffiffiffiffiffiffiffiffi
3 3
64 ¼ 4 because ð 4Þ ¼ð 4Þð 4Þð 4Þ¼ 64:
If n is even, b is assumed to be the nonnegative root. Also even roots of
negative numbers do not exist in the real number system. In this book, it is
assumed that even roots will be taken only of nonnegative numbers. For
p ffiffiffi
instance in x, it is assumed that x is not negative.
Root properties are similar to exponent properties.
p ffiffiffiffiffi p p
n
ffiffiffi ffiffiffi
Property 1 n ab ¼ n a b
We can take the product then the root or take the individual roots then the
product.

Examples

p ffiffiffiffiffi p ffiffiffiffiffiffiffiffiffiffiffi p ffiffiffi p ffiffiffiffiffi
64 ¼ 4 16 ¼ 4 16 ¼ 2 4 ¼ 8
p ffiffiffi ffiffiffiffiffiffip p ffiffiffiffiffiffiffiffi p ffiffiffiffiffiffi ffiffiffiffiffi p ffiffiffiffiffiffiffiffiffiffi
p
5 5 5 4 4 4
3 4x ¼ 12x 6x 4y ¼ 24xy
Property 1 only applies to multiplication. There is no similar property for
addition (nor subtraction). A common mistake is to ‘‘simplify’’ the sum of
p ffiffiffiffiffiffiffiffiffiffiffiffiffi
2
two squares. For example x þ 9 ¼ x þ 3 is incorrect. The following
example should give you an idea of why these two expressions are not
p ffiffiffiffiffiffiffiffiffiffiffi p p ffiffiffi
equal. If there were the property n a þ b ¼ n ffiffiffi n b, then we would have
a þ
p ffiffiffiffiffi p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ffiffiffiffiffi p ffiffiffi
58 ¼ 49 þ 9 ¼ 49 þ 9 ¼ 7 þ 3 ¼ 10:
2
This could only be true if 10 = 58.
p
r ffiffiffi ffiffiffi
a n a
n
Property 2 ¼ p ffiffiffi
b n b
We can take the quotient then the root or the individual roots then the
quotient.
p
r ffiffiffi ffiffiffi
4 4 2
¼ p ¼
9 9 3
ffiffiffi
p p
ffiffiffi m n ffiffiffiffiffiffi
m
Property 3 n a ¼ a (Remember that if n is even, then a must not be
negative.)
We can take the root then the power or the power then take the root.