What Are Exponents in Math?

Exponents are the opposites of roots.
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Exponents in math are typically superscript numbers or variables written beside another number or variable. Exponentiation is any mathematical operation which uses exponents. Each form of exponent must follow unique rules in order to be solved; in addition, some exponential forms are central to real life rules and applications.

Notation

The notation of an exponent in math is a pair of numbers, symbols or both. The number written normally is called the base number, while the number written in superscript is the exponent. The root form of most exponents is a number multiplied with itself by the exponent’s number of times. For example, the notation 5 x 5 x 5 is the root form of the exponentiation, 5 raised to 3, sometimes written as 5^3.

Order of Operation

In the order of operations, PEMDAS, solving exponents is second order. Exponents are resolved after all equations in parenthesis have been completed, but before doing any multiplication and division. Complex exponential notations act as equations in themselves and must be solved first prior to the primary equation.

Notable Exponents

Math uses specific terminology for some common exponents. The term “squared” is used for numbers raised to the power of 2. “Cubed” is used for numbers raised to the power of 3. Other exponents have particular rules for them. For example, a number raised to 1 is itself and any number raised to 0, except 0, is always 1.

Basic Rules: Addition/Subtraction

In algebra, both variables have to have the same base and exponent to be added or subtracted. For example while x^2 added to x^2results to 2x^2, x^2 added to x^3 cannot be solved as is. To solve these types of equations, each exponent has to be factored out until both variables are in their base form or have the same exponent.

Basic Rules: Multiplication/Division

In algebra, if the same variable with different exponents are multiplied or divided against each other, the exponents add or subtract themselves respectively. For example, x^2 multiplied by x^2 would equal x^4. X^3 divided by x^2 would equal x^1, or simply, x. Additionally, an exponential is divided by itself if it has a negative exponent. For example, x^-2 would result in 1 divided by x^2.

Applications

Exponents have been used in multiple scientific applications. For example, half-life is an exponential notation which states how many years a compound has before it reaches half of its lifespan. It is also used in business as well; stock prices are estimated by using exponential growth rates based on historical data. Lastly, it has daily life implications as well. Most driving schools warn drivers about the implications of speeding: if the car speed is simply doubled, the braking distance is typically multiplied by an exponential factor.

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