EXAMPLE 13 Reducing the Index of a Radical
Simplify:
.
SOLUTION

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Check Point
13
Simplify:
.
Blitzer Bonus || A Radical Idea: Time Is Relative
The Persistence of Memory
(1931), Salvador Dali:. © 2011 MOMA/ARS.
What does travel in space have to do with radicals? Imagine that in the future we will be able to travel at velocities approaching the speed of light
(approximately 186,000 miles per second). According to Einstein’s theory of special relativity, time would pass more quickly on Earth than it would in the
moving spaceship. The special-relativity equation
gives the aging rate of an astronaut,
R
a
, relative to the aging rate of a friend,
R
f
, on Earth. In this formula,
v
is the astronaut’s speed and
c
is the speed of light.
As the astronaut’s speed approaches the speed of light, we can substitute
c
for
v
.
Close to the speed of light, the astronaut’s aging rate,
R
a
, relative to a friend,
R
f
, on Earth is nearly 0. What does this mean? As we age here on Earth, the space
traveler would barely get older. The space traveler would return to an unknown futuristic world in which friends and loved ones would be long gone.
CONCEPT AND VOCABULARY CHECK
Fill in each blank so that the resulting statement is true.

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1.
The symbol is used to denote the nonnegative, or __________, square root of a number.
2.
because _____ = 64.
3.
_____
4.
The product rule for square roots states that if
a
and
b
are nonnegative, then
= _____.
5.
The quotient rule for square roots states that if
a
and
b
are nonnegative and
= _____.
6.
= _____
7.
___
= _____
8.
The conjugate of 7 +
is __________.
9.
We rationalize the denominator of
by multiplying the numerator and denominator by __________.
10.
In the expression
, the number 3 is called the _____ and the number 64 is called the __________.
11.
because _____ = −32.
12.
If
n
is odd,
= _____.
If
n
is even,
= _____.
13.
= _____
14.
= (___)
3
= ___
EXERCISE SET P.3
Practice Exercises
Evaluate each expression in Exercises 1–12, or indicate that the root is not a real number.
1.

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2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
Use the product rule to simplify the expressions in Exercises 13–22. In Exercises 17–22, assume that variables represent nonnegative real numbers.
13.
14.
15.
16.
17.
18.

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19.
20.
21.
22.
Use the quotient rule to simplify the expressions in Exercises 23–32. Assume that x
> 0.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.

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In Exercises 33–44, add or subtract terms whenever possible.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
In Exercises 45–54, rationalize the denominator.
45.
46.
47.
48.

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49.

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