Download Exponents are shorthand for repeated multiplication and

Exponents are shorthand for repeated multiplication and can be used
to express an n-th root of a number: b is a number x such that xn= b.
LEARNING OBJECTIVE [ edit ]
Evaluate exponential expressions of the form , , and
KEY POINTS [ edit ]
The number in larger font is called the base. The number in superscript (that is, the smaller
number written above) is called the exponent.
If b is a positive real number and n is a positive integer, then there is exactly one positive real
solution to xn = b. This solution is called the principal n-th root of b. It is denoted n√b, where √ is
the radical symbol; alternatively, it may be written b1/n.
A power of a positive real number b with a rational exponent m/n in lowest terms satisfies m
b
n
= (b
m
1
)
n
n
‾‾‾ .
= √b
m
TERM [ edit ]
rational number
A real number that can be expressed as the ratio of two integers.
Give us feedback on this content: FULL TEXT [edit ]
Exponentsare a shorthand used for repeated multiplication. Remember that when you were
first introduced to multiplication it was as a shorthand for repeated addition. For example,
you learned that: 4 ∗ 5
= 5 + 5 + 5 + 5
. Theexpression "4 × " told us how many times we
needed to add. Exponents are the same type of shorthand for multiplication. Exponents are
written in superscript after a regular-sized number.
For example: 2
3
= 2
∗ 2 ∗ 2 . The number in larger font is called the base. The number in
superscript (that is, the smaller number
written above) is called the exponent. The
exponent tells us how many times the base
is multiplied by itself. In this example, 2 is
the base and 3 is the exponent.The
expression 23is read aloud as "2 raised to
the third power", or simply "2 cubed".
Here are some other examples: 6
∗ 6 = 6 (This would read aloud as "six
2
times six is six raised to the second power"
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or more simply "six times six is six
squared". ) 7 ∗ 7 ∗ 7 ∗ 7
4
= 7
(This would read aloud as "seven times seven times seven
times seven equals seven raised to the fourth power". There are no alternate expression for
raised to the fourth power. It is only the second and third powers that usually get abbreviated
because they come up more often. When it is clear what is being talked about, people often
drop the words "raised" and "power" and might simply say "seven to the fourth". )
Rational Exponents
A rational exponent is a rational number that can be used as another way to write roots. An
n-th root of a number b is a number x such that xn = b.
If b is a positive real number and n is a positive integer, then there is exactly one positive real
solution to x
n
= b
. This solution is called the principal n-th root of b. It is denoted n√b,
1
1
n
2
where √ is the radical symbol; alternatively, it may be written b . For example: 4
1
83 = 2
= 2
, .
When one speaks of the n-th root of a positive real number b, one usually means the principal
n-th root.
m
b
n
m
= (b
1
)n
n
‾‾‾
m
= √b
where m is an integer and n is a positive integer. Rational powers m/n, where m/n is in lowest terms, are positive if m is even, negative for
negative b if m and n are odd, and can be either sign if b is positive and n is even. 1
(27) 3 =
−3 , (27)
2
3
1
=
−9 , and 64 has two roots 8 and −8. Since there is no real number x
2
such that x2 = −1, the definition of bm/n when b is negative and n is even must use
the imaginaryunit i. A power of a positive real number b with a rational exponent m/n in
lowest terms satisfies:
If n is even, then xn = b has two real solutions; if b is positive, which are the positive and negative nth roots. The equation has no solution in real numbers if b is negative. If n is odd, then xn = b has one real solution. The solution is positive if b is positive and negative if b is negative.
Examples of exponents graphed can be seen in this figure .
Interactive Graph: Exponential graph with different substitute "b" values
Graphs of y
= b
x
for various bases (b): 10 (red), e (blue), 2 (green), and ½ (purple). Each curve passes
through the point (0, 1) because any nonzero number raised to the power of 0 is 1. At x
= 1
, the y­value
equals the base because any number raised to the power of 1 is the number itself. Most notable is y = (
1
2
)
x
. Why does it look different than the other graphs?