Page 115 - MarceAlgebra Demystified

102 CHAPTER 5 Exponents and Roots

p ffiffiffi p p p
r ffiffiffi ffiffiffi ffiffiffiffiffi ffiffiffiffiffi
6 6 7 42 42
3: ¼ p p ¼ p ffiffiffiffiffi ¼
7 ffiffiffi ffiffiffi 7 2 7
7
7
p ffiffiffi p ffiffiffi p ffiffiffi
8x 8x 3 8x 3 8x 3
4: p ¼ p p ¼ p ffiffiffiffiffi ¼
ffiffiffi
ffiffiffi
ffiffiffi
3 3 3 3 2 3
p ffiffiffiffiffiffiffiffi p p p
r ffiffiffiffiffiffiffiffi ffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffi
7xy 7xy 11 77xy 77xy
5: ¼ p ffiffiffiffiffi p ffiffiffiffiffi ¼ p ffiffiffiffiffiffiffi ¼
11 11 11 11 2 11
In the case of a cube (or higher) root, multiplying the fraction by
the denominator over itself usually does not work. To eliminate the nth
root in the denominator, we need to write the denominator as the
nth root of some quantity to the nth power. For example, to simplify
1 3
p ffiffiffi we need a 5 under the cube root sign. There is only one 5 under the
3
5
cube root. We need a total of three 5s, so we need two more 5s. Multiply
2
3
5by5 to get 5 :
p ffiffiffiffiffi p ffiffiffiffiffi p ffiffiffiffiffi
1 3 5 2 3 5 2 3 25
p ffiffiffi p ffiffiffiffiffi ¼ p ffiffiffiffiffi ¼ :
3 3 2 3 3
5 5 5 5
When the denominator is written as a power (often the power is 1) sub-
tract this power from the root. The factor will have this number as a
power.
Examples

8 The root minus the power 4 3 ¼ 1:
p
1
4 ffiffiffiffiffi We need another x under the root.
3
x
p ffiffiffiffiffi p p
ffiffiffi
8 4 x 1 8 x 8 x
4
4
ffiffiffi
p ffiffiffiffiffi p ffiffiffiffiffi ¼ p ffiffiffiffiffi ¼
4 3 4 1 4 4 x
x x x
4x The root minus the power is 5 2 ¼ 3:
p ffiffiffiffiffi
3
5 2 We need another x under the root.
x
p ffiffiffiffiffi p ffiffiffiffiffi p ffiffiffiffiffi
5
5
4x 5 x 3 4x x 3 4x x 3 p ffiffiffiffiffi
5
p ffiffiffiffiffi p ffiffiffiffiffi ¼ p ffiffiffiffiffi ¼ ¼ 4 x 3
5 2 5 3 5 5 x
x x x
p ffiffiffi p p p
r ffiffiffiffiffi 7 7 ffiffiffiffiffi 4 7 ffiffiffiffiffiffiffi 4 7 ffiffiffiffiffiffiffi 4
2 2 x 2x 2x
7
¼ p ffiffiffiffiffi p ffiffiffiffiffi ¼ p ffiffiffiffiffi ¼
x 3 7 x 3 4 x 4 7 x 7 x