Exponents are used to showing repeated multiplication of a number by itself. Writing large numbers sometimes becomes tedious. In large mathematical expressions, they occupy more space and take more time. This issue is resolved by the use of exponents. For example, 7 × 7 × 7 can be represented as
What is Meant by Integers as Exponents?
In Mathematic, the integers exponents are the exponents that should be an integer. It can be either a positive integer or a negative integer. In this, the positive integer exponents describe how many times the base number should be multiplied by itself. Whereas the negative integer exponents first describe flipping the numerator and the denominator value and define to multiply the number by itself for the number of times mentioned there.
Exponents Vs Powers
We know that the expression 6 x 6 can be calculated, but the expression can also be written in a short manner that is known as exponents.
6.6 = 6^2
The expression that describes repetitive multiplication of same value is known as power. The value 6 is known as base or power and the number 2 is known as an exponent. It corresponds to the number of times the base is operated as a factor.
Integer Exponent Rules
| Rule name | Rule | Example |
| Product rule | a n· am = an+m | 23 · 24 = 23+4 = 128 |
| a n · b n = (a · b) n | 32 · 42= (3·4)2 = 144 | |
| Quotient rule | a n/ a m = an-m | 25 / 23 = 25-3 = 4 |
| a n / b n = (a / b)n | 43 / 23= (4/2)3 = 8 | |
| Power rule | (bn)m = bnm | (23)2 = 23.2 = 64 |
| m√(bn) = bn/m | 2√(26) = 26/2 = 8 | |
| b1/n = n√b | 81/3 = 3√8 = 2 | |
| Negative exponent | b-n= 1 / bn | 2-3 = 1/23 = 0.125 |
| Zero rule | b0 = 1 | 60 = 1 |
| 0n = 0 , for n>0 | 06= 0 | |
| One rule | b1 = b | 71 = 7 |
| 1n = 1, n= even number | 18 = 1 | |
| Minus one rule | (-1)n= -1 , n= odd number | (-1)5 = -1 |
| Derivative rule | (xn)’ = n·x n-1 | (x3)’ = 3·x3-1 |
| Integral rule | ∫ xndx = xn+1/(n+1)+C | ∫ x2dx = x2+1/(2+1)+C |
Integers with Positive and Negative Exponents
We know that
=>
=>
So,
Similarly,
In general we can say that for any non-zero integer say ‘a’ ,
Integers as Exponents Examples
Example 1:
Find the multiplicative inverse of
Solution:
Therefore the multiplicative inverse of
Example 2:
Find the multiplicative inverse of
Solution:
The multiplicative inverse of
Example 3:
Expand the number 12345 in the exponent form.
Solution:
The number 12345 can be expressed as:
12345 = 1 × 10000 + 2 × 1000 + 3 × 100 + 4 × 10 + 5 × 1
=> 12345 = 1 ×
Similarly, this method can be employed to decimal numbers also.
Example 4:
Expand the number 987.65 in the exponent form.
Solution:
The number 987.65 can be represented as:
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