(d)
The greatest perfect cubethat divides into 16 is 8, so factor 16 as
8 is a perfect cube.Product rule= 2232 238 = 2
= 238 # 232
= 238 # 2
2316
8 #2.
2316
446 CHAPTER 8 Roots, Radicals, and Root Functions
Remember to
write the index.(e)
81 is a perfect 4th power.Product rule=- 3242 2481 = 3 NOW TRY
=- 2481 # 242
=- 2481 # 2
- 24162
Remember the
negative sign
in each line.CAUTION Be careful with which factors belong outside the radical sign and
which belong inside.Note in Example 4(b)how is written outside because
and while the remaining 3 is left inside the radical.
Simplifying Radicals Involving VariablesSimplify. Assume that all variables represent positive real numbers.
(a)
Factor.Product ruleTake the square root.Absolute value bars are not needed around the min color because all the variables
represent positivereal numbers.
(b)
Factor.Remove perfect square factors.(c)
Choose as the perfect cube that
divides into.
Product ruleTake the cube root.(d)
is the greatest 4th power that dividesProduct rule=- 2 y^2242 y Take the fourth root. NOW TRY
=- 2416 y^8 # 242 y
=- 24116 y^8212 y 2 16 y^832 y^9.
- 2432 y^9
= - 2 xy 3 xy^2
= 23 - 8 x^3 y^3 # 23 xy^2
- 8 x^4 y^5
- 8 x^3 y^3
= 231 - 8 x^3 y^321 xy^22
23 - 8 x^4 y^5
= 10 k^3 q^422 k
= 2102 # 2 # 1 k^322 #k# 1 q^422
2200 k^7 q^8
= 4 m 2 m
= 216 m^2 # 2 m
= 216 m^2 #m
216 m^3
EXAMPLE 5
222 = 2 232 =3,
2 # 3
NOW TRY
EXERCISE 4
Simplify.
(a) (b)
(c) (d)
(e)- 2480
242 23108
250 2192
NOW TRY ANSWERS
- (a) (b)
(c) cannot be simplified
further.
(d) 3234 (e)- 2245
242522 823NOW TRY
EXERCISE 5
Simplify. Assume that all
variables represent positive
real numbers.
(a) (b)
(c)
(d) - 24162 x^7 y^8
23 - 125 k^3 p^7236 x^5232 m^5 n^4- (a) (b)
(c)- 5 kp^223 p (d) - 3 xy^2242 x^3
6 x^22 x 4 m^2 n^222 m