2
6a
MODULE 6. RADICAL EXPRESSIONS
Square Roots
Introduction to Radical Notation
We know how to square a number. For example:
• 52 = 25
• (−5)2 = 25
Taking the square root of a number is the opposite of squaring.
• The nonnegative square root of 25 is 5.
• The negative square root of 25 is −5.
Thus, when searching for a square root of a number, we are searching for
number whose square is equal to our number.
You Try It!
EXAMPLE 1. Find the square roots of 81.
Solution: We are looking for a number whose square is 81.
• Because 92 = 81, the nonnegative square root of 81 is 9.
• Because (−9)2 = 81, the negative square root of 81 is −9.
Hence, 81 has two square roots, −9 and 9.
!
You Try It!
EXAMPLE 2. Find the square roots of 0.
Solution: We are looking for a number whose square is 0.
• Because 02 = 0, the nonnegative square root of 0 is 0.
No other number squared will equal zero. Hence, zero has exactly one square
root, namely zero.
!
You Try It!
EXAMPLE 3. Find the square roots of −36.
3
6A. SQUARE ROOTS
Solution: We are looking for a number whose square is −36. However, every
time you square a real number, the result is never negative. Hence, −36 has
no real square roots.1
!
The introductions in Examples 1, 2, and 3 lead to the following definition.
Defining the square roots of a number. The solutions of x2 = a are called
square roots of a.
√
Case: a > 0. The equation x2 = a has two real solutions, namely x = ± a.
√
• The notation a calls for the nonegative square root.
√
• The notation − a calls for the egative square root.
Case: a = 0. The equation x2 = a has exactly one solution, namely x = 0.
Case: a < 0. The equation x2 = a has no real solutions.
You Try It!
√
√
EXAMPLE 4. Solve x2 = 9 for x, then simplify 9 and − 9.
Solution: There are two numbers whose square equal 9, namely −3 and 3.
Hence:
x2 = 9
x = ±3
Original equation.
Two answers: (−3)2 = 9 and 32 = 9.
Writing x = ±3 is a shortcut for writing x = −3 or x = 3. Next:
√
• 9 calls for the nonnegative square root of 9. Hence:
√
9=3
√
• − 9 calls for the negative square root of 9. Hence:
√
− 9 = −3
!
1 When we say that −36 has no real square roots, we mean there are no real numbers
that are square roots of −36. The reason we emphasize the word real in this situation is
the fact there −36 does have two square roots that are elements of the complex numbers,
a set of numbers that are usually introduced in advanced courses such as college algebra or
trigonometry.
4
MODULE 6. RADICAL EXPRESSIONS
You Try It!
EXAMPLE 5. Solve x2 = 0 for x, then simplify
√
0.
Solution: There is only one number whose square equals 0, namely 0. Hence:
x2 = 0
x=0
Original equation.
One answer: (0)2 = 0.
Thus, the only solution of x2 = 0 is x = 0. Next:
√
• 0 calls for the nonnegative square root of 0. Hence:
√
0=0
!
You Try It!
EXAMPLE 6. Solve x2 = −4 for x, then simplify
√
−4.
Solution: You cannot square a real number and get negative. Hence, x2 = −4
has no real solutions. Further:
√
• −4 calls for the nonnegative square root of −4. Because you cannot
square a real number and get −4, there is no nonegative square root of
−4.
!
You Try It!
EXAMPLE 7. Simplify each of the following:
√
√
√
√
b) − 225
c) −100
d) − 324
a) 121
√
Solution: Remember, the√notation a calls for the nonnegative square root
of a, while the notation − a calls for the negative square root of a.
• Because 112 = 121, the nonnegative square root of 121 is 11. Thus:
√
121 = 11
• Because (−15)2 = 225, the negative square root of 225 is −15. Thus:
√
− 225 = −15
• You cannot square a real number and get −100. Therefore,
a real number.
√
−100 is not
5
6A. SQUARE ROOTS
• Because (−18)2 = 324, the negative square root of 324 is −18. Thus:
√
− 324 = −18
!
Squaring “undos” taking the square root.
√
√
Squaring square roots. If a > 0, then both − a and a are solutions of
x2 = a. Consequently, if we substitute each of them into the equation x2 = a,
we get:
! √ "2
!√ "2
− a =a
and
a =a
You Try It!
EXAMPLE 8. Simplify each of the following expressions:
√
√
√
a) ( 5)2
b) (− 7)2
c) ( −11)2
Solution: We’ll handle each case carefully.
√
√
a) Because 5 is a solution of x2 = 5, if we square 5, we should get 5.
√
( 5)2 = 5
√
√
b) Because − 7 is a solution of x2 = 7, if we square − 7, we should get 7.
√
(− 7)2 = 7
√
c) Because x2 = −11 has no real answers, −11 is not a real number. Advanced courses such as college algebra or trigonometry will introduce the
complex number system√and show how to handle this expression. At this
level, we comment that −11 is not a real number and cease and desist.
!
Approximating Square Roots
The squares in the “List of Squares” are called perfect squares. Each is the
square of a whole
√ number. Not all numbers are perfect squares. For example,
in the case of 24, there is no√whole number whose square is equal to 24.
However, this does not prevent 24 from being a perfectly good number.
We can use the “List of Squares” to find decimal approximations when the
radicand is not a perfect square.
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MODULE 6. RADICAL EXPRESSIONS
You Try It!
Estimate:
√
83
List of Squares
x
x2
0
0
1
1
2
4
3
9
4
16
5
25
6
36
7
49
8
64
9
81
10
100
11
121
12
144
13
169
14
196
15
225
16
256
17
289
18
324
19
361
20
400
21
441
22
484
23
529
24
576
25
625
EXAMPLE√9.
Estimate 24 by guessing.
Solution. From the “List of Squares,”
√ note that 24 lies betwen 16 and 25, so
√
24 will lie between 4 and 5, with 24 much closer to 5 than it is to 4.
√
16
√
24
√
25
4
4.8
5
Let’s guess
√
24 ≈ 4.8.
As a check, let’s square 4.8.
(4.8)2 = (4.8)(4.8) = 23.04
Not quite 24! Clearly,
√
24 must be a little bit bigger than 4.8.
√
Just for fun, here is a decimal approximation of 24 that is accurate to
1000 places, courtesy of http://www.wolframalpha.com/.
4.8989794855663561963945681494117827839318949613133402568653851
3450192075491463005307971886620928046963718920245322837824971773
09196755146832515679024745571056578254950553531424952602105418235
40446962621357973381707264886705091208067617617878749171135693149
44872260828854054043234840367660016317961567602617940145738798726
16743161888016008874773750983290293078782900240894528962666325870
21889483627026570990088932343453262850995296636249008023132090729
18018687172335863967331332533818263813071727532210516312358732472
35822058934417670915102576710597966482011173804100128309322482347
06798820862115985796934679065105574720836593103436607820735600767
24633259464660565809954782094852720141025275395093777354012819859
11851434656929005776183028851492605205905926474151050068455119830
90852562596006129344159884850604575685241068135895720093193879959
87119508123342717309306912496416512553772738561882612744867017729
60314496926744648947590909762887695867274018394820295570465751182
126319692156620734019070649453
If you were to multiply this number by itself (square the number), you would
get a number that is extremely close to 24, but it would not be exactly 24.
There would still be a little discrepancy.
!
Answer: 9.1