1) Multiply and simplify by factoring assume that all expressions under radicals represent
nonnegative numbers ^3sqrt y^4 ^3sqrt 16y^5
y 4 3 16y 5 = 2y 3 3 2
3
2) simplify by factoring assume that all expressions under radicals represent nonnegative
! numbers ^4sqrt 80x^12y^14
4
80x12 y14 = 2x 3 y 3 4 5y 2
3) simplify by factoring assume that all expressions under radicals represent nonnegative
! numbers ^5sqrt 160x^7y^15
5
!
160x 7 y15 = 2xy 3 5 5x 2
4) multiply and simplify by factoring sqrt125 times sqrt90
125 90 = 75 2
5) multiply and simplify by factoring assume that all expressions under radicals represent
! nonnegative numbers sqrt5b^7 sqrt10c^8
5b 7 10c 8 = 5b 3c 4 2b
6) divide then simplify by taking roots if possible assume that all expressions under
! radicals represent positive numbers sqrt 35a/sqrt 5a
35a
= 7
5a
7) divide then simplify by taking roots if possible assume that all expressions under
! radicals represent positive numbers ^3sqrt 54a^4b^8/^3sqrt 2a^2b^7
3
54a 4 b 8
3
!
2 7
2a b
= 33 a 2b
8) simplify by taking roots of the numerator and denominator assume that all expressions
under radicals represent positive numbers ^5 sqrt 243x^7/y^10
243x 7 3x 5 2
= 2 x
y10
y
5
9) add or subtract simplify by collecting like radical terms if possible 9sqrt3! 6sqrt3+9sqrt3
9 3 +6 3 "9 3 =6 3
!
10) add simplify by collecting like radical terms if possible 7sqrt75+8sqrt27
7 75 + 8 27 = 59 3
11) add simplify by collecting like radical terms if possible assuming that all expressions
! under radicals represent nonnegative numbers sqrt2a+^5sqrt8a^3
2a + 5 8a 3 = (1+10a) 2a
!
12) multiply sqrt11 (5-4sqrt11)
(
)
11 5 " 4 11 = 5 11 " 44
!
13) multiply ^3sqrt a (^3sqrt5a^2+^3sqrt625a^2)
3
!
a
(
3
)
5a 2 + a 625a 2 = 6a 3 5
14) multiply (2+sqrt2)(2-sqrt2)
(2 + 2 )(2 " 2 ) = 2
!
15) multiply (^5sqrt9-^5sqrt3)(^5sqrt8+^5sqrt27)
(
!
5
9 "5 3
)(
5
)
8 + 5 27 = 3 " 5 81 + 5 72 " 5 24
16) rationalize the denominator 6sqrt7/7sqrt3
6 7 2 21
=
7
7 3
17) rationalize the denominator assume that all expressions under radicals represent
! positive numbers ^3sqrt2y^4/^3sqrt6x^4
3
2y 4
3
4
6x
=
y 3 2
9x y
3x 2
18) rationalize the denominator assume that all expressions under radicals represent
! positive numbers 4-sqrt a/2+sqrt a
4 " a a "6 a +8
=
4"a
2+ a
19) rationalize the denominator assume that all expressions under radicals represent
! positive numbers sqrt c-sqrt d/sqrt c+sqrt d
c " d c " 2 cd + d
=
c "d
c+ d
!