4.1 Introduction: An exponent is a simple notation used for multiplying that number of identical factors. Exponentiation is mathematical operation written in the form of a^n where n is an integer and a is any value.
an = a x a x a x……. a
a is called the base and n is called the exponent.
example: a=4; n=2
an = 42 = 4 x 4 = 16
an is read as a powered n or a raised to the power n.
4.2.1 Positive integer exponents:
n= positive integer
a2 = a x a is called the square of a
a3 = a x a x a is called the cube of a
The word "raised" is usually not used neither is the word "power" used so 35 is typically pronounced "three to the fifth" or "three to the five.
4.2.2 Negative integer exponents: A negative exponent means to divide by that number of factors instead of multiplying. So 3-3 is the same as 1/(33), therefore in general term we can write it as
a-n = (1/an)
example: 5-3 can be written as
5-3= (1/ 53) = 1/(5x5x5) = 1/125
Note* n≠ 0, n = 0 is not defined for a-n
4.2.3 Exponents of zero and one: We know that anythingto the power 0 is 1.
Assume a0. By the division rule we know that,
an/an = a(n-n) = a0. (i)
But anything divided by itself is 1, so
If an/an is equal to 1 and from (i) we have an/an equals to a0, then 1 must equal a0. Symbolically,
an/an = a(n-n) = a0 = 1
we created a fraction to figure out a0. But division by 0 is not allowed, so our evaluation is defined for anything to the 0 power except zero itself.
4.3.1 Powers of ten: In the decimal number system, integer powers of 10 are written as the digit 1 followed by a number of zeroes, depending on the sign and magnitude of the exponent.
For example, 103 = 1000
10-4 = 0.0001.
For example: 3452178 can be written as
34.52178 x 105
SI prefixes based on powers of 10 are also used to describe various small and large quantities.
the prefix kilo means 103 = 1000, so
1 km= 1000m.
1 kg = 1000gm
In the base 2 (binary) number system, integer powers of 2 are written as 1 followed or preceded by a number of zeroes determined by the sign and magnitude of the exponent. For example, two to the power of three is written 1000 in binary.
4.3.3 Powers of one: The integer powers of one are one: 1n = 1.
4.3.4 Powers of zero: If the exponent is positive, the power of zero is zero: 0n = 0, where n > 0.
If the exponent is negative, the power of zero (0n, where n < 0) is undefined, because division by zero is implied.
4.3.5 Powers of minus one: If n is an even integer, then (-1)n = 1. If n is an odd integer, then (-1)n = −1. Because of this, powers of −1 are useful for expressing alternating sequences.
i) am+n = am . an
This identity has the consequence
ii) am-n = am = 1
a = an-m
for a ≠ 0,
iii) (am)n = am.n
Another basic identity is
iv) (a.b)m = am . bm
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