Download A quick overview of roots

NOTES: A quick overview of roots
[Make sure you’ve studied the overview of exponents handout first.]
radical sign
3
√25
square root of 25
√8
cube root of 8
LEVEL 1
REVIEW: What’s a perfect square?
9 is a perfect square since 32 = 9
the result of squaring a whole number
25 is a perfect square since 52 = 25
The GED Testing Service recommends memorizing the 1st 12 perfect squares:
12 = 1
22 = 4
32 = 9
42 = 16
52 = 25
62 = 36
72 = 49
82 = 64
92 = 81
102 = 100
112 = 121
122 = 144
How do I find the square root of a number? (*)
Just as multiplication and division are opposites, squaring and square rooting
are opposites. To find the square root, you do squaring “backwards.”
42 = 16 (4 squared = 16) and √16 = 4 (the square root of 16 is 4)
52 = 25 (5 squared = 25) and √25 = 5 (the square root of 25 is 5)
To begin, practice finding the square root of perfect squares.
Ask: what number times itself equals the number inside the radical sign (√ ) ?
What’s the square root of 9?
√9 = ?
_____  _____ = 9 [What single number can you put in both the blanks?]
3 is the square root of 9.
(*) ADVANCED NOTE: 3 times itself also equals 9. So 9 actually has 2 square roots: 3 and 3. However, most
applications involve positive square root answers, and we’ll focus only on those. Further, strictly speaking, the √
symbol means the principal square root, that is, the positive one. If we want to indicate the negative square root, we’d
need to write √ . Just as we don’t bother writing +3 to indicate positive 3, we don’t bother writing + √ . The positive
value is assumed when there’s no sign in front. So the technically correct answer to “What is √9 ?” is just 3, not 3 (as
some books & web sites misleadingly claim). For the negative square root we’d need to ask “What is √9 ?” and the
answer there is 3. Notice we’re not taking the square root of a negative number; that’s undefined for real numbers.
What we’re doing is indicating the negative square root of positive 9. To avoid unnecessary complication, I’ll just say
“square root” instead of “principal square root,” but I’ll always mean the positive one. [If this note doesn’t make sense,
you may safely ignore it. ]
D. Stark 12/8/2016
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For the square root of fractions, find the square roots of the numerator and
4
denominator separately: √ =
9
√4
√9
=
2
3
The square root of a negative number is undefined in the world of real
numbers: √9 = 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑
Can you figure out why?
REVIEW: What’s a perfect cube?
3
8 is a perfect cube since 2 = 8
the result of cubing a whole number
64 is a perfect cube since 43 = 64
The GED Testing Service recommends memorizing the 1st 6 perfect cubes:
13 = 1
23 = 8
33 = 27
43 = 64
53 = 125
63 = 216
How do I find the cube root of a number?
Think of cube rooting as cubing “backwards.”
To begin, practice finding the cube root of perfect cubes.
3
Ask: what number times itself times itself equals the number inside the √
?
3
What’s the cube root of 8?
√8 =?
_____  _____  _____ = 8 [What single # can you put in all the blanks?]
2 is the cube root of 8.
For the cube root of fractions, find the cube roots of the numerator and
3
denominator separately: √
64
125
3
=
√64
3
√125
=
4
5
Until you study imaginary numbers, the square root of a negative number
is undefined: √9 = 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑 But we can find the cube root of
3
negative numbers. √– 8 = 2
Why is there this difference?
When do we encounter roots?
 when working “backwards” from area or volume to a side length or radius
 when doing the last step of Pythagorean Theorem (a2 + b2 = c2) problems—
finding a or b or c instead of just a2 or b2 or c2
[If c2 = 25 then c = √25 = 5]
 when using the quadratic formula (found on your formula sheet):
−𝑏 ± √𝑏 2 − 4𝑎𝑐
𝑥=
2𝑎
D. Stark 12/8/2016
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LEVEL 2
Simplifying non-perfect roots:
To simplify a root that isn’t a perfect square, see if you can find a factor that is a
perfect square. Then split the root into 2 parts & simplify the part you can.
EXAMPLE: √12 = √4  3 = √4  √3 = 2  √3 = 2√3
Just as we don’t need a multiplication sign between a coefficient and a variable,
we don’t need one between the whole number part and the remaining root.
EXAMPLE: √500 = √100  5 = √100  √5 = 10  √5 = 10√5
More on the relation of squaring and square rooting:
 Multiplying and dividing “undo” one another. If you multiply a number by 10 and
then divide by 10, you’re back to the starting number.
 Squaring and finding the square root work the same way.
o
If you square a number and then take the square root,
then you’re back to where you started.
Start with 5. Square it: 52 Square root that: √52 = √25 = 5.
o
If you square root a number and then square that,
then you’re back to where you began.
2
Start with 5. Square root it: √5 Square that: (√5)
2
(√5) = (√5)(√5) = √5  5 = √25 = 5
 Be able to recognize that √42 = 4
√32 = 3 etc.
LEVEL 3
Simplifying roots with variables:
[likely beyond the scope of the GED test]
perfect roots: √𝑥 8 = √𝑥 4  𝑥 4 = 𝑥 4
√25𝑦 6 = √25  𝑦 3  𝑦 3 = √25  √𝑦 3  𝑦 3 = 5𝑦 3
non-perfect roots: √𝑥 4 𝑦 7 = √𝑥 2  𝑥 2  𝑦 3  𝑦 3  𝑦1
= √𝑥 2  𝑥 2  √𝑦 3  𝑦 3  √𝑦1 = 𝑥 2 𝑦 3 √y
GED® is a registered trademark of the American Council on Education (ACE) and administered exclusively by GED
Testing Service LLC under license. This material is not endorsed or approved by ACE or GED Testing Service.
D. Stark 12/8/2016
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