Rational exponents can be defined as an exponent that is written as a fraction. It represents both an integer exponent and the nth root. Nth root is the number that must be multiplied n times by itself to equal a given value. It can be written as nx or xn1. The base is the x while the exponent is 1n.

In any situation, the exponent is always applied first while rewriting any equation followed by the radical, but if the base is negative, taking roots is no simpler, but it requires complex number exponentiation.

They are real numbers that can be expressed either while writing a definition essay as a ratio of 2 finite integers:

- x = L/M, L ∈ Z, M ∈ Z

Applying the property 2 of exponents, we have:

- ax = aL/M = (a1/M)L
- Thus, we have a1/M. Since (a1/M)M = aM/M = a. We see that a1/M is the Mth root of a.

The Mth root of a real number is not unique. As we all know, any square root gives two values, i.e., 4 = ±2. In the general case of Mth roots, there are M distinct values in general.

Some examples are given below:

- 532 = 2
- It means 25 = 32
- Where n = 2 is a nth root and is called a square root. Also, if n is even and x negative, then nx is nonreal.

The root is found in the denominator i.e. at the bottom while the integer exponent is found in the numerator i.e. on top. Another example is explained below:

- Rule: am/n = nam or (na )m
- 52/3 = 352 = 325 165/4 = (416 )5 = 25 = 32

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## Main Approaches To Rational Exponents As Used In The Mathematical World

There are many rules used in rational exponents, and they include product rules, power rules for quotient, power rules for a product, zero, negative and rational exponents, power rule, quotient rule, one rule, minus one rule, derivative rule, and integral rule.

The product rule if the base (a) of any two numbers in multiplication are the same and their exponents (n, m) are different, and the exponent will be multiplied while their bases remain the same. For instance, if a, a are positive real numbers while their exponents n is any real number, then we have something like this;

- an • am = an+m
- 23 • 24 = 23+4 = 27 or (128) OR 23 x 24 = (2x2x2) x (2x2x2x2) = 27 or (128)

When the exponents of any two numbers in multiplication are the same, then their bases will be multiplied while their exponents remain the same. For instance, if a, b (different bases) are positive real numbers but their exponent n is any real number, then we have something like this:

- an • bn = (a • b)n
- 32 • 42 = (3 • 4)2 = 122 or (144)

Power rules for quotient: when the bases of two numbers in the division are the same, the exponents will be subtracted while the base remains the same. If a is a positive real number and n, m, are any real numbers, then we have:

- an / am = an-m
- Therefore; 25 / 23 = 25-3 = 22. This can also be written as; 2 • 2 = 4

Power rules for quotients have other sub-rules:

- Quotient rule with the same base
- Quotient rule with the same exponent
- Power rule I
- Power rule II
- Power rule with radicals

Quotient rule with the same base can be written as: an / am = an-m. Examples include:

- 25 / 23 = 25-3 = 22

Quotient rule with the same exponents can be written as: an / bn = (a / b)n. Example:

- 43 / 23 = (4/2)3 = 23 = 2·2·2 = 8

Power rule I is written as (an) m = a n·m. Example:

- (23)2 = 23·2 = 26 = 2·2·2·2·2·2 = 64

Power rule II can be written as: a nm = a (nm). Example:

- 232 = 2(32) = 2(3·3) = 29 = 2·2·2·2·2·2·2·2·2 = 512

Power rule with radicals can be written as: m√(a n) = a n/m. Example:

- 2√(26) = 26/2 = 23 = 2·2·2 = 8

Negative exponents: for any real number a, where a≠0, a-n=1/an. The fraction 1/an is called the reciprocal of an. Below are set of examples where the negative exponent rules are being used:

- Example 1: What is 2-3 =? This can be written as: 2-3 = 1/23 = 1/8
- Example 2: What is 35 ÷ 37 =? Also rewritten as: 35-7 = 3-2 = 1/32 = 1/9
- Example 3: Simplify 4-3/16-1. Rewritten as: 4-3/(42)-1 = 4-3/4—2 = 4-3-(-2) = 4-1 = ¼

Zero exponent rule: this rule states that any base b raised to the power of zero (0) equals one (1). Also written as follows: 50 = 1.

### Theories and Researches Made at Solving Tasks

The law of exponents has been discovered long before now by an ancient Greek mathematician, Archimedes. He proved that: 10a x 10b = 10a+b, is important to manipulate the powers of 10. Muhammad Ibn Musa, in the 9th century, used the term mal for a square and kab for a cube, which was later represented in mathematical notation as m and k, respectively.

Jost Burgi used the Roman numerals for exponents in the late 16th century. Early in the 17th century, the very first form of the modern exponential notation was introduced by Rene Descartes. The word exponent was finally coined in the year 1544 by Michael Stifel. Some mathematicians such as Isaac Newton used exponents only for powers larger than two, preferring to represent squares multiplication. Thus, in order to formulate polynomials while writing a synthesis essay, it should be written like this: ax + bxx + cx3 + d.

In 1748, Leonard Euler wrote considering exponentials in which the exponent is a variable itself. It is clear that quantities of this kind cannot be algebraic functions since the exponents must be constant with this introduction of transcendental functions.

### Modern Situation to Rational Exponents and the World of Technology

Rational exponents, indices, index numbers, and powers are all used in our modern-day world of advanced technology. Credits to those great researchers and mathematicians who discovered and made APA essays after solving exponents thereby, making the world understand that the use of them is not just in mathematics anymore, but in every aspect of life and science as well in general.

Computer Games, Physics, Science, Engineering, Accounting, pH, and Richter Measuring Scales, Finance, and Economics, among others, are those areas where the use of rational exponents is much essential.

Computer games use game physics engines of a low-level program within the game. These engines with a great thesis methodology are used to calculate the precise movement, interactions, and also geometry involved with the game. A lot of Algebra formulas in their Algorithms are being put to use for these games to work and function as expected by the end consumers, i.e., the players. If the mathematical theories used in these game engines are not correct, they will fail to work. Other fields where exponential growths are applied include:

- Demographics
- Biology
- Resources
- Electronics

People who use rational exponents are Economists, Bankers, Biologists, Financial Advisors, Insurance Risk Assessors, Chemists, Engineers, Sound Engineers, Statisticians, Computer Programmers, Physicists, Geographers, Mathematicians, Geologists as well as many other professions.

Exponential Decay – if we use a negative power value, results in infractions when these fractions have exponents applied to them, we get a Decay. Furthermore, in a Decay process, the amount that is involved drops off quickly at the beginning, but then slower and slower goes the drop-off towards the end.

Exponential Growth - the equations for graphs of these rational exponents situations contain some exponents, and these results in the graph starting off slow but then increasing very rapidly. e.g., Think of square numbers and how these numbers become bigger and bigger: e.g., 1 4 9 16 25 36 49 64 81 100 121 132, etc.

Exponents are also part of Food Technology and Microbiology – just like in viral marketing where a business plan is being used and passed from one person to another before they meet their targets, illnesses arising from virus infections, including many emails as well as computer viruses, are able to spread at an ever-increasing exponential rate causing a major widespread to the infected area. A more real-life example is the missile launch and bomb explosions where there is an uncontrolled massive increasing output of energy and force within a very short period of time. Imagine this as a very steep exponential graph, compared to a match burning in a fairly flat straight line graph giving out energy.

Exponential Scales - the Richter Scale is used to evaluate the severeness of earthquakes. The actual energy from each quake that occurs at any given time period is a power of 10, but on the scale, we only take the index value of 1, 2, 3, 4, rather than the full rational exponents quantity. This means that: Richter Scale of 6 earthquakes will be 10 times stronger than a Richter Scale of 5 earthquakes. (for example: 1000000 vs 100000). A Richter Scale of 7 earthquakes is 100 times stronger than a Richter Scale of 5 earthquakes. (e.g. 20000000 vs 100000). The pH scale used in the measurement of the acidity of any material is also created by taking the values of power from the measured powers of 10 acid concentration values.

Exponents and scientific notation - very large numbers such as the distance between planets or the population of a particular country can be expressed using powers of 10 in a format called scientific notation. Scientific notation can be used for representing very minute decimal values like the size of flu virus molecules or the distance between atoms in a crystal structure.