Exponents
Exponents are termed expressions that consist of a repeated power of multiplication of the identical factor. It is also known as raised to the power of indices or a number.
For example, the number 64 can be written as 4^3.
In the given expression the index/power is 3 and the base is 4. The value of the expression is calculated by multiplying the base multiple times as the number of power, thus the value of the above expression becomes 4 x 4 x 4 = 64.
Types of Exponents
Based on the nature of the number in the power, exponents can be divided into four major types:
● Positive Exponents
Positive exponents can be simply calculated just by multiplying the base to itself the number of times indicated by the index/power.
● Negative Exponents
Negative exponents can be calculated by placing 1 in the numerator and the base along with the exponent in the denominator of a fraction.
● Zero Exponents
Zero exponents in this the base value is ignored and any expression with the exponent as 0 is equivalent to 1.
● Rational Exponents
Rational or Fractional exponents will become roots or radical. For example, 3^(⅓) can be written as 3 root of 3, 8^(5/2) can be written as 2 root (or square root) of 8 raises to the power 5.
Exponential Function
An exponential function is a mathematical function, which is mainly used to find the exponential decay or exponential growth in real-world scenarios for example to compute investments, model populations, and so on.
An exponential function is a standard mathematical function in form f(x) = ax, where “x” is a variable and “a” is a constant, called the base of the function.
The most commonly used exponential function base is Euler’s number “e”, which is approximately equal to 2.71828.
Exponential Function Formula
The exponential function is a standard mathematical function which is of the form
f(x) = ax
Where
x is a real number.
a>0 and a not equal to 1
If the variable is negative, the function is indefinite for -1<x<1.
Here,
“x” is a variable
“a” is a constant, which is the base of the function.
The exponential function is responsible for the exponential curve’s growth or decay. Any quantity that grows or decays by a specified percent at periodic intervals should retain either exponential growth or exponential decay.
Exponential Growth
In Exponential Growth, the quantity increases very gradually initially, and then rapidly. The rate of change grows over time. The rate of growth increases with time. The rapid growth is meant to be an “exponential increase”.
The formula to define exponential growth is:
y = a(1+ r)x
Where r is the growth percentage.
Exponential Decay
In Exponential Decay, the quantity decreases very rapidly initially, and then gradually. The rate of change falls over time. The rate of change becomes slower with time. The rapid decay is meant to be an “exponential decrease”.
The formula to define exponential growth is:
y = a(1- r)x
Where r is the decay percentage.
Laws of Exponents and Exponential Functions
The rules of exponents are used to solve many mathematical problems which involve repeated multiplication operations. The laws of exponents and exponential function simplify the multiplication and division operations and help to solve the problems effortlessly.
Following are the laws:
| Product law | am.an = am+n |
| Quotient law | am/an = am-n |
| Power raised to a power | (am)n = amn |
| Product to a power | an.bn = (ab)n |
| Quotient to a power | an/bn = (a/b)n |
| Zero power | a0 = 1 |
| Negative exponent | a-m = 1/am |
| Fraction exponent | a1/n = n√a |
Examples
Example 1: Which one has the greater value 5^3 or 3^5?
Solution:
The value of,
5^3 = 5 x 5 x 5 = 125 and,
3^5 = 3 x 3 x 3 x 3 x 3 = 243.
Therefore, 5^3 < 3^5
Example 2:What is the value of 50 + 22 + 40 + 71 – 31 ?
Solution:
50 + 22 + 40 + 71 – 31 = 1+4+1+7-3= 10
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