Exponentiation
Exponentiation is a fundamental mathematical operation that involves raising a base to an exponent, which indicates how many times the base is multiplied by itself. The base can be a natural number or a negative integer, while the exponent is often represented as a superscript. This operation is also known as involution and is integral to various mathematical concepts, including exponential equations and scientific notation.
Historically, the roots of exponentiation can be traced back to ancient civilizations, with significant contributions from mathematicians like Michael Stifel in the 16th century, who formalized exponent notation. Key concepts related to exponentiation include rules such as the product rule, power rule, and zero rule, which govern how to manipulate exponential expressions.
Additionally, fractional exponents provide a way to express roots, linking radical expressions to their exponential counterparts. Understanding exponentiation is crucial for solving problems in various fields, including mathematics, science, and engineering, where exponential growth and decay are commonly encountered. Overall, exponentiation serves as a vital building block in both theoretical and applied mathematics.
Exponentiation
Exponentiation, a mathematical operation, corresponds to multiplication. Written with a base and an exponent. The base can be either a natural number or a negative integer. The exponent shows how many times to multiply the base with itself. Exponentiation is also called involution because you are raising a number to a power.
In this article, the history of exponentiation and various terms such as "like terms", associative property, radical numbers, integers, coefficients, and commutative properties will all be discussed. Exponent rules, exponential equations, exponential models, fractional exponents, exponential expression, and simplifying expressions will all be discussed in detail.
Overview
The prefix "expo" is derived from the Latin language, meaning place. The idea of "squaring" dates back to Babylonian times and a tablet from the twenty-third century BC that had examples of squaring numbers. The first appearance of exponents in literature was in the book Arithemetica Integra (1544), written by the German mathematician Michael Stifel. Stifel was simply working with the base two and the exponent three to get eight.
Michael Stifel was born in 1497 in Esslingen, Germany and died in Jena, Germany in 1567. He attended the University of Wittenberg where he received his Master’s Degree after attempting to use methods of numerology to uncover religious meanings and secrets. Michael became a serious mathematician when he decided to settle down in Holzdorf, Germany. While residing there, he wrote Arthhmetica Intergra, Deustsche Arithmetica, and Welsche Practick. These books were major contributions to mathematics and exponential notation.
Logarithms were introduced by Scottish baron John Napier and Swiss craftsman Joost Burgi in the 1500’s. They produced independent systems that embodied logarithmic relations. Thomas Robert Malthus is best known for his work "An Essay on the Principle of Population" (1798). The essay predicts that the world will be unable to feed itself because of the exponentially increasing population.
Key terms to know when using exponentiation are associative property, radical number, integer, coefficient, like term, commutative property, and exponent. "Like" terms are terms that have the same variable and powers; the coefficients do not have to match. Associative property relates to grouping numbers. Coefficient is a number used to multiply a variable. The commutative property states that the order of number does not matter and applies only to multiplication and addition. Exponent is a number that represents the power, and exponents are usually expressed as a raised number next to another number or expression. A radical number is an expression that has a square root, cube root and so on. An integer is a whole number and a number that is not a fraction.
Exponent Rules
There are several rules for exponents. The first rule is that any number multiplied by the power of one equals itself, for example, x1 = x or 31 = 3. The second rule, which is referred to as the product rule, states that when multiplying two powers that have the same base, the exponents can be added together as a shortcut. The expression xm × xn, for example, is the same as xm+n. Likewise, 42+3 = 45.
The power rule, states that to raise a power to a power, multiply the exponents together. For example, (52)3 = 52×3 = 56. The quotient rule, states two powers can be divided with the same base by subtracting the exponents. For example,
The zero rule states that any number multiplied by the power zero equals to one (x0 = 1). The rule of the negative exponents, states that any number raised to a negative power equals to its reciprocal raised to the opposite positive number.
Exponential Equations
An exponential equation is one in which a variable occurs in the exponent. In an exponential equation where both of the bases are the same, set the exponents equal in order to solve for x. When the equation cannot be expressed in terms of a common base. It will be necessary to use a logarithm to find a solution. A logarithm is an exponent to which the base must be multiplied to produce that number. For example, the logarithm of a thousand to base ten is three. Because ten to the power three is one thousand.
To solve most exponential equations, first isolate the exponential expression. The second step is to take log, or ln, of both sides. Lastly, solve for the variable and check. The exponential equation 72x+1 = 73x−2 does not require a logarithm because the terms express a common base. The equation 5x = 7, however, does require a logarithm because the bases are not common.
Fractional Exponents
Fractional exponents are also referred to as rational exponents and are an alternate way to express roots. When dealing with fractional exponents, the denominator of the rational exponent becomes the index of the radical. Then the numerator becomes the expression inside the radical (as in
) , which is also referred to as the radicand. A radical expression can be converted to an expression containing a fractional power, which makes solving the problem easier.
There are relationships when dealing with fractional exponents. For the square root, you can write it as one-half power. The cube is written as one-third power. The fourth root is written one-fourth power. The fifth root is written as one-fifth root, and so on. French mathematician Albert Girard was the first to actually express fractional exponents.
Exponential Expression
The use of an exponential expression is a way to show the repeated multiplication of a number itself. When the exponent is two, it is referred to as squaring the number. If the exponent is a three, then it is referred to as cubing the term. If the number is extremely large or extremely small, you can use scientific notation to shorten it up using exponents. In order to add exponential terms, the base and the exponent must be the same. In order to multiply the exponents, both of the bases must be the same. Lastly, when dividing exponents, the bases be the same (like multiplying exponents) and the exponents are subtracted from another.
Simplifying Expression with Exponents
When simplifying an expression with exponents, it is important to follow the rules of exponents and the rules of addition, subtraction, and multiplication. First, simplify each term according to multiplication, division, and power-to-power rules. Next, combine like terms and arrange them according to which have variables and which have the highest exponent. In some cases like terms are only evident in using the power-to-power rule. For example,
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