Table of Contents
Laws of Exponent
The law of exponent, or laws of powers, are a set of three mathematical laws that describe the behavior of exponential functions. The laws are
1. The law of addition: The product of two powers is the sum of the two powers.
For example, the product of 3 and 4 is 12.
3 × 4 = (3 + 4) = 12
2. The law of multiplication: The power of a number is the product of the number and the power.
For example, the power of 5 is 25.
5 × 5 = 25
3. The law of exponentiation: The power of a power is the product of the two powers.
For example, the power of 3 raised to the power of 4 is 81.
3 × 3 × 3 × 3 = (3 × 4) = 81
What is an Exponent?
An exponent is a number that is written above and to the right of a base number and that indicates how many times the base number is to be multiplied by itself.
1. Product of Power or Product Law
The product of two powers is the product of the individual powers multiplied together. For example, the product of 3 and 4 is 12.
2. Quotient of Power or Quotient Law
The quotient of power, also called the quotient law, is a mathematical law that states that the power of a number divided by another number is equal to the power of the second number divided by the first number. In mathematical notation, this law can be expressed as follows:
\(P/Q = (Q^2)/(P^2) = Q\)
This law is important in the study of exponents, as it allows for the simplification of expressions that involve division of powers.
3. Power of Power or Power Law
The power of a power or power law is a mathematical concept that describes the behavior of a quantity that is proportional to a power of another quantity. In other words, a power law describes a relationship between two variables in which one variable is a constant multiple of the other.
The mathematical formula for a power law is: y = kx^n, where y is the variable that is proportional to x, k is the constant multiplier, and n is the power.
The power law is used to model a wide variety of physical and mathematical phenomena, including the distribution of income and wealth, the sizes of cities, the frequencies of words in a language, and the properties of crystals.
4. Pareto Principle
The Pareto principle or Pareto’s law is a principle that states that, for many events, roughly 80% of the effects come from 20% of the causes.
The Pareto principle was first observed by Italian economist Vilfredo Pareto in the late 1800s, who found that 80% of the land in Italy was owned by 20% of the population.
The principle has been applied to a wide variety of areas, including business, engineering, and ecology.
4. Power of a Product Law
The product law is a powerful tool in the hands of a patentee. The law allows a patentee to prevent others from making, using, or selling the patented product without the patentee’s permission.
5. Patentability Requirements
In order to be patentable, an invention must meet a number of requirements. The invention must be new, useful, and non-obvious.
5. Power of a Quotient Law Theorem
The power of a quotient law states that for any real numbers a and b,
(a/b) raised to the power c is equal to ac/bc.
Proof
We will use the property of exponents to prove this theorem.
Let a/b be equal to x and c be any real number.
Then, x raised to the power c is equal to (a/b) raised to the power c, or ac/bc.
Negative Power Law
The negative power law is a mathematical law that states that the number of objects in a system decreases exponentially as the power of the objects decreases.
The negative power law is also known as the negative binomial distribution, the negative power law of decay, and the geometric distribution.
Fractional Power Law
The fractional power law is a mathematical relationship between two variables, typically the size of a population and the average size of its members. The fractional power law states that the average size of a population is inversely proportional to the square of the population size.
This law can be used to model the size distribution of a population of organisms. The average size of the population decreases as the population size increases, and the population size approaches zero, the average size approaches infinity.
