Rational and Radical Expressions
1. Simplify the rational expression:
x² – 6x – 7
———————————
x² – 1
Factor top and bottom:
(x - 7)(x + 1)
————————————————
(x - 1)(x + 1)
Cancel the (x + 1)'s
1
(x - 7)(x + 1)
————————————————
(x - 1)(x + 1)
1
and all that's left is
x - 7
———————
x - 1
2. Divide:
x² – 3x + 2 x² – 4
————————————— ÷ ——————————
8x – 8 5x + 10
Invert the second fraction and change ÷ to ·
x² – 3x + 2 5x + 10
————————————— · ——————————
8x – 8 x² - 4
Factor all numerators and denominators:
(x - 2)(x - 1) 5(x + 2)
———————————————— · ————————————————
8(x – 1) (x - 2)(x + 2)
Indicate the multiplication of all factors
in the numerator and denominator all as
one fraction:
5(x - 2)(x - 1)(x + 2)
——————————————————————————
8(x – 1)(x - 2)(x + 2)
Cancel the (x - 2)'s
1
5(x - 2)(x - 1)(x + 2)
——————————————————————————
8(x – 1)(x - 2)(x + 2)
1
1 1
5(x - 2)(x - 1)(x + 2)
——————————————————————————
8(x – 1)(x - 2)(x + 2)
1 1
Cancel the (x + 2)'s
1 1 1
5(x - 2)(x - 1)(x + 2)
——————————————————————————
8(x – 1)(x - 2)(x + 2)
1 1 1
all that's left is
5
———
8
3. Simplify:
__________
√100x2y16z8
Write the 10 as 102
To take the square root of an even power,
divide the exponent by 2:
__________
√102x22y16z8
102÷2x2÷2y16÷2z8÷2
101x1y8z4
Erase the two 1 exponents
10xy8z4
4. Perform the indicated operations:
__ __ __
Ö72 + Ö32 – Ö18
72 = 8·9 = 4·2·3·3 = 2·2·2·3·3
32 = 4·8 = 2·2·4·2 = 2·2·2·2·2
18 = 2·9 = 2·3·3
_________ _________ _____
Ö2·2·2·3·3 + Ö2·2·2·2·2 - Ö2·3·3
Group like factor into pairs:
_____________ _____________ _______
Ö(2·2)·2·(3·3) + Ö(2·2)·(2·2)·2 - Ö2·(3·3)
Each pair of like factors comes out in front
of the square root radical as a single factor.
That is, in the first radical the (2·2) and
the (3·3) come out in front of the radical as
as single factors multiplied 2·3, and the
unpaired 2 stays under the radical.
_ _ _
2·3Ö2 + 2·2Ö2 - 3Ö2
_ _ _
6Ö2 + 4Ö2 - 3Ö2
_
Factor out Ö2
_
Ö2(6 + 4 - 3)
_
Ö2(7)
_
7Ö2
5. Multiply:
_ _ _ _
(Ö3 + Ö5)(2Ö3 – Ö5)
Use FOIL
_ _ _ _ _ _ _ _
(Ö3)(2Ö3) – (Ö3)(Ö5) + (Ö5)(2Ö3) – (Ö5)(Ö5)
Multply under the radicals:
___ ___ ___ ___
2Ö3·3 - Ö3·5 + 2Ö5·3 - Ö5·5
Pair like factors under the 1st and 4th radicals
Multiply factors under the middle two radicals
which have no pairs of like factors:
_____ __ __ _____
2Ö(3·3) - Ö15 + 2Ö15 - Ö(5·5)
The pair of 3's comes out eliminating the radical
in the first term. The middle two terms combine,
and the pair of 5's comes out eliminating the
radical in the 4th term:
__
2·3 + Ö15 - 5
__
6 + Ö15 - 5
__
1 + Ö15
6. Rationalize the denominator:
3
———————————
Ö6 – Ö3
Form the two term conjugate of the denominator.
1. Its first term is_the same as the first term of
the denominator Ö6
2. Its second term is the second term of the _
denominator with its sign changed, or_Ö6 + Ö3
3. So the conjugate of Ö6 - Ö3 is Ö6 + Ö3
4. Put the conjugate over itself
_ _
Ö6 + Ö3
———————————
Ö6 + Ö3
which just equals 1, so we can then multiply the
fraction to be rationalized by this fraction without
changing its value, since multiplication by 1 does
no change the value of an expression
_ _
3 Ö6 + Ö3
——————————— · ———————————
Ö6 – Ö3 Ö6 + Ö3
Indicate the multiplication of numerators and
denominators, all as one fraction:
_ _
3(Ö6 + Ö3)
————————————————————
(Ö6 – Ö3)(Ö6 + Ö3)
FOIL out the bottom:
_ _
3(Ö6 + Ö3)
———————————————————————————
Ö6Ö6 + Ö3Ö6 - Ö6Ö3 - Ö3Ö3
_ _
3(Ö6 + Ö3)
———————————————————————
Ö36 + Ö18 - Ö18 - Ö9
_ _
3(Ö6 + Ö3)
———————————————————————
6 + Ö18 - Ö18 - 3
The middle terms cancel
_ _
3(Ö6 + Ö3)
————————————
6 - 3
_ _
3(Ö6 + Ö3)
————————————
3
Cancel the 3's
1 _ _
3(Ö6 + Ö3)
————————————
3
1
_ _
Ö6 + Ö3
Edwin
