1.06 Radicals
and Fractional Exponents
College Algebra: One Step at a Time,
Page
66-76: #38, 39, 41, 42, 43, 45, 53, 67, 70, 80, 88, 89, 90.
Dr. Robert J. Rapalje
Seminole State College of Florida
Altamonte Springs Campus
To see
Section 1.06, with
detailed explanations, examples, exercises, and answers,
click here!
First, before you even start to do
#38, there are cube roots, so you must get the perfect cubes in your mind:
23=8,
33=27,
43=64,
53=125.
38.
5
− 4
Find a perfect cube that divides into
108
(that would be
27) and a perfect cube that divides into
32
(that would be
8).
108=27x4
and 32=8x4.
5
− 4
5
− 4
5
− 4
5 3
− 4
2
Now multiply the numbers 5
times 3
and the
4
times 2.
15 −
8
Now you have like terms so you can combine the 15
and the -8, and the final answer
is 7
In # 39, you have 4th roots, so keep in mind that
24=16
and 34=81.
39.
7 
−
3 
Find a perfect 4th power that divides
into 32
(that would be
16) and a perfect 4th power
that divides into 162
(that would be
81).
32=16x2
and 162=81x2.
7
− 3
7
−
3
7
−
3
7 2
−
3 3
Now multiply the numbers 7
times
2
and the
3
times
3.
14
9
Now you have like terms so you can combine the 14
and the -9, and the final answer is
41.
First, separate each of the square roots into two square roots. Sort out
the square roots into perfect squares that go in the
first (red)
square root, and the left-over factors that
go in the second (blue) square root.
Everyone can take the square root of the
first (red) radicals
since they are perfect squares. Nobody knows what to do about the
second (blue) radical
since they cannot be simplified. So do what you can do
(the red radicals),
and leave the
rest (blue radicals!)
alone:
As in
the last step, you do what you are able to do next--multiply outside the
radicals:
Notice
that what is left turn out to be like radicals and like terms. They combine
together:
42. 
First, separate each of the square roots into two square roots. Sort out
the square roots into perfect squares that go in the
first (red)
square root, and the left-over factors that
go in the second (blue) square root.
Everyone can take the square root of the
first (red) radicals
since they are perfect squares. Nobody knows what to do about the
second (blue) radical
since they cannot be simplified. So do what you can do
(the red radicals),
and leave the rest
(blue radicals!)
alone:
As in
the last step, you do what you are able to do next--multiply outside the
radicals:
Notice
that what is left turn out to be like radicals and like terms. They combine
together:
43.
First, separate each of the cube roots into two cube roots. Sort out the
factors that are perfect cubes and place them in the
first (red)
cube root, and place any left-over factors
in the second (blue) cube root.
Everyone can take the cube root of the
first (red) radicals
since they are perfect
cubes. Nobody knows what to do about the
second (blue) radical
since they cannot be simplified. So do what you can do
(the red radicals),
and leave the rest
(blue radicals!)
alone:
As in
the last step, you do what you are able to do next--multiply outside the
radicals:
Notice
that what is left turn out to be like radicals and like terms. They combine
together:
45.
First, separate each of the square roots into two square roots. Sort out
the perfect squares, and place these in the
first (red)
square root, and put the left-over factors
in the second (blue) square root.
Everyone can take the square root of the
first (red) radicals since they are perfect
squares. Nobody knows what to do about the
second (blue) radical since they cannot be
simplified. So do what you can do (the red
radicals), and leave the rest
(blue radicals!)
alone:
As in
the last step, you do what you are able to do next--multiply outside the
radicals:
Notice
that what is left turn out to be like radicals. It subtracts out, the final
answer is
53.
Since
there are no obvious perfect cube factors in this problem, use the product
property of radicals to multiply 12 times 6. In this case, the numbers are
small enough to just perform the multiplication.
The
perfect cube that divides into 72 is 8, so this can be written with the
perfect cube first:
Since this is a numerical problem, you can check the answer by
calculating the value of the problem:
= 4.160167646 . . .
and compare it to the decimal value of the answer that you
obtained:
= 4.160167646 . . .
67.
Remember, you multiply the numbers that are
OUTSIDE the radical together, and you keep
them OUTSIDE
the radical. Then you multiply the numbers that are
INSIDE the radical
together and keep them INSIDE
the radical.
Now,
simplify the radical 45. Break it down into 9 times 5.
FINAL ANSWER!
Since
this is a numerical problem, you can check the answer by calculating the
value of the problem:
= 160.9968944 . . .
and
compare it to the decimal value of the answer that you obtained:
= 160.9968944 . . .
70.
Remember, you multiply the numbers that are
OUTSIDE the radical together, and you keep
them OUTSIDE
the radical. Then you multiply the numbers that are
INSIDE the radical together and keep
them INSIDE
the radical.
However, if you use a calculator and multiply out the numbers that are
INSIDE
the radical, you end up with a large number that you wont know how to
simplify. Its better, instead of multiplying the numbers out, to break
them down into prime factors, and for square roots, look for pairs of
numbers, for cube roots, look for three of a kind, etc.
Notice
that you have three
factors of
5!
That makes a perfect cube:
Now,
separate into two radicals, with the perfect cube in the first, and the
leftover factors in the second radical.
FINAL ANSWER!!
Since
this is a numerical problem, it is a be a good place to check the answer by
calculating the problem:
= 236.9996855 . . .
and
then calculate the answer that you obtained:
= 236.9996855 . . .
(Note:
as I was working this problem for you, I made a mistake and I missed it!
Because of this check, I had to go back and find an error of my own!!)
80.
F
O I
L
As a
check, calculate the value of the problem:
= -150.6217783
Then, calculate the value of the answer: = -150.6217783
88.
Multiply the
first (5)
times everything in the second parentheses:
Next, multiply the
second (
) times everything in the
second parentheses:
Now, put it ALL together and
combine like terms:
=
=
=
=
89.
First, multiply the second binomial times the third binomial:
Now,
F
O I L
this out and simplify the result:
F
O I
L
= _____________________
≈
0.471890505
≈
0.471890505.
90.
First, multiply the second binomial times the third binomial:
Now,
F
O I
L
this out and simplify the
result:
F
O I
L
= _____________________
≈ 6288.817045
≈ 6288.817045 .
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