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Powers, Exponents, and Roots
Squares, Cubes, and Higher Order Exponents
Squares
The square of a number is that number times itself. 5 squared, denoted
52, is equal to 5×5, or 25. 2 squared is 22 = 2×2 = 4.
One way to remember the term "square" is that there are two dimensions in a
square (height and width) and the number being squared appears twice in
the calculation. In fact, the term "square" is no coincidence--the square of a
number is the area of the square with sides equal to that number.
A number that is the square of a whole number is called a perfect square.
42 = 16, so 16 is a perfect square. 25 and 4 are also perfect
squares. We can list the perfect squares in order, starting with 12:
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, ...
Cubes
The cube of a number is that number times itself times itself. 5 cubed,
denoted 53, is equal to 5×5×5, or 125. 2 cubed is
23 = 2×2×2 = 8. The term "cube" can be remembered
because there are three dimensions in a cube (height, width, and depth)
and the number being cubed appears three times in the calculation.
Similar to the square, the cube of a number is the volume of the cube
with sides equal to that number--this will come in handy in higher levels of math.
Exponents
The "2" in "52" and the "3" in "53" are called
exponents. An exponent indicates the number of times we must multiply the
base number. To compute 52, we multiply 5 two times (5×5), and
to compute 53, we multiply 5 three times (5×5×5).
Exponents can be greater than 2 or 3. In fact, an exponent can be any number.
We write an expression such as "74" and say "seven to the fourth
power." Similarly, 59 is "five to the ninth power," and
1156 is "eleven to the fifty-sixth power."
Since any number times zero is zero, zero to any (positive) power is always
zero. For example, 031 = 0.
A number to the first power is that number one time, or simply that number: for
example, 61 = 6 and 531 = 53. We define a number to
the zero power as 1: 80 = 1, (- 17)0 = 1, and
5210 = 1.
Here is a list of the powers of two:
| 20
|
= |
1 |
|
| 21
|
= |
2 |
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| 22
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= |
2×2 = 4 |
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| 23
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= |
2×2×2 = 8 |
|
| 24
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= |
2×2×2×2 = 16 |
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| 25
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= |
2×2×2×2×2 = 32 |
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and so on...
Exponents and the Base Ten System
Here is a list of the powers of ten:
| 100
|
= |
1 |
|
| 101
|
= |
10 |
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| 102
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= |
10×10 = 100 |
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| 103
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= |
10×10×10 = 1, 000 |
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| 104
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= |
10×10×10×10 = 10, 000 |
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| 105
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= |
10×10×10×10×10 = 100, 000 |
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and so on...
Look familiar? 100 is 1 one (a 1 in the ones place), 101
is 1 ten (a 1 in the tens place), 102 is 1 hundred, 103 is
1 thousand, 104 is 1 ten thousand, etc. This is the meaning of
base ten--a "1" in each place represents a
number in which the base is 10 and the exponent is the number of zeros after the
1. The place value is the number that is multiplied by this number. For
example, a 5 in the thousands place is equivalent to 5×1000, or
5×103.
We can write out any number as a sum of single-digit numbers times powers of
ten. The number 492 has a 4 in the hundreds place (4×102), a 9 in
the tens place (9×101) and a 2 in the ones place (2×100).
Thus, 492 = 4×102 +9×101 +2×100.
Examples: Write out the following numbers as single-digit numbers times
powers of ten.
935 = 9×102 +3×101 +5×100
67, 128 = 6×104 +7×103 +1×102 +2×101 +8×100
4, 040 = 4×103 +0×102 +4×101 +0×100
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