Download A quick overview of basic exponents and roots

NOTES: A quick overview of basic exponents & roots
Exponents:
power
exponent
[base with exponent]
[Sometimes the exponent alone is also called the power.]
104
base
The exponent says how many times the base is multiplied by itself.
101 = 10
[ten to the 1st power]
102 = 10  10 = 100
[ten to the 2nd power; ten squared]
103 = 10  10  10 = 1,000
[ten to the 3rd power; ten cubed]
104 = 10  10  10  10 = 10,000
[ten to the 4th power]
Any number to the 1st power is just the number itself: 21 = 2 51 = 5 11 = 1
How do I square a number?
42 = 4  4 = 16
01 = 0
find the number times itself
[NOTE: Don’t multiply the 4 and the 2.
The exponent (2) just says how many instances of the base (4) are multiplied.]
62 = 6  6 = 36
We call it squaring since the result is the area of a square.
A square that is 6 inches on each side has an area of 36 square inches.
.52 = .5  .5 = .25
1 2
( ) =
12
22
=
1
You can square decimals, fractions, and integers, too.
1 2
OR ( ) =
1

1
=
1
2
4
2
2
2 4
[You can treat numerator & denominator separately or think of the whole fraction times itself.]
Be careful! (3)2 is not the same as 32
(3)2 = 3  3 = 9
32 = (3  3) = 9
The exponent goes with the base that’s right next to it. The parentheses enclose the negative
number and make the exponent apply to the whole thing. Without the parentheses, the exponent
applies only to the number alone without the negative sign, which just gets applied at the end.
If you want to square a negative number on your calculator, make sure to use parentheses!
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Basic exponents & roots
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How do I cube a number?
find the number times itself times itself
43 = 4  4  4 = 64
53 = 5  5  5 = 125
We call it cubing since the result is the volume of a cube.
A cube that is 5 inches on each side has a volume of 125 cubic inches.
.63 = .6  .6  .6 = .216
1 3
( ) =
13
=
1
1 3
OR ( ) =
You can cube decimals, fractions, and integers, too.
1

1

1
=
1
2
23
8
2
2
2
2
8
[You can treat numerator & denominator separately or think of the whole fraction cubed.]
Notice that (2)3 is the same as 23. Both equal 8.
So there’s a difference between having and not having parentheses with
negative numbers for squaring but not for cubing. Can you figure out why?
When do we encounter exponents?
 when calculating area—We use squaring and square units (sq. ft, sq. in.)
o figuring out the size of a room to paint
o determining how much carpet you need
 when calculating volume—We use cubing and cubic units (cu. ft, cu. in.)
o finding how much room volume a heater or air conditioner will deal with
 when using the Pythagorean Theorem (a2 + b2 = c2) to calculate side lengths of
a right triangle
o telling the length of a diagonal walkway for a garden
 when using quadratic equations, which can model the path of a ball or a missile,
the shape of satellite dish, and much, much more
o Here’s the quadratic formula (found on your formula sheet):
−𝑏 ± √𝑏 2 − 4𝑎𝑐
𝑥=
2𝑎
D. Stark 3/22/2017
Basic exponents & roots
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Roots:
radical sign
3
√25
√8
square root of 25
What’s a perfect square?
cube root of 8
the result of squaring a whole number
9 is a perfect square since 32 = 9
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.
For the square root of fractions, find the square roots of the numerator and
4
denominator separately: √ =
9
√4
√9
=
2
3
(*) 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 3/22/2017
Basic exponents & roots
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The square root of a negative number is undefined in the world of real
numbers: √9 = 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑
Can you figure out why?
What’s a perfect cube?
the result of cubing a whole number
8 is a perfect cube since 2 = 8
64 is a perfect cube since 43 = 64
3
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
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.
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Basic exponents & roots
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