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Understanding the components of this mathematical expression is key to grasping the various exponent rules. Dream01 / ShutterstockWorking with exponents can cause numbers to grow extremely large or shrink to minuscule sizes almost instantly. Having a few handy shortcuts can make this process much easier.
Once you've mastered the exponent rules, also known as the laws of exponents, you can use them to solve mathematical problems that would otherwise require more time and effort.
What Is an Exponent?
An exponent represents a base value a raised to the power n, expressed as a. In simple terms, it signifies a number multiplied by itself a specific number of times. Exponents, like many mathematical concepts, serve as a symbolic notation to simplify the representation of equations.
For example, consider 2 raised to the power of 3 (also called 2 cubed; the exponent three is "cubed," and the exponent two is "squared"). The calculation would proceed as follows:
Here, two is the base number (or base value), and three is the exponent (or exponent value). The base is the larger number in standard font, while the exponent is the smaller number written in superscript.
Thus, 2 to the power of 3 — often referred to as "two to the third power" — equals 8.
Why Are Exponent Rules Helpful?
As you might imagine, especially if you have some background in mathematics, situations can become complex very quickly. For instance, raising a number to the eighth power can result in extremely large figures. Additionally, you might encounter negative numbers raised to powers, numbers raised to negative exponents, fractional exponents, and more.
This is where exponent rules prove invaluable. By understanding these rules or keeping a handy exponent rules chart, you can simplify the calculations needed to arrive at the correct answer. Mastering the appropriate exponent rule (or power rule) for a given equation not only saves time but also ensures accuracy.
8 Essential Exponent Rules With Examples
Here are eight crucial exponent rules to master. Whether you use a chart or create one yourself, having this reference readily available can be incredibly useful.
The Product Rule
This rule explains that when multiplying two exponential expressions with the same base, you can add their exponents and then raise the base to this combined exponent. Symbolically, the product law of exponents is represented as: a x a = a.
Example:
The Quotient Rule
Often referred to as the quotient law, this rule explains that when dividing two expressions with the same base, you can subtract the exponents and then raise the base to the resulting power. Symbolically, the quotient rule of exponents is expressed as: (a)/(a) = a.
Example:
The Zero Exponent Rule
Also referred to as the zero power rule, this principle states that any number raised to the power of zero equals 1. The only exception is zero raised to the zero power, which is considered an indeterminate form. Symbolically, the zero exponent law is represented as: a = 1.
Example:
The Identity Exponent Rule
The identity law states that any number raised to the power of 1 remains unchanged. Symbolically, this rule is expressed as: a = a.
Example:
The Negative Exponent Rule
This rule states that any number raised to a negative exponent should be solved by taking its reciprocal. The base and exponent are moved to the denominator, with a one in the numerator, and the exponent's sign is changed to positive.
Symbolically, the negative exponent law is represented as: a = 1/(a).
Example:
The Power of a Power Rule
Symbolically, the power of a power law is expressed as: (a^n)^m = a^nm.
Example:
The Power of a Product Rule
When a product is raised to an exponent, the exponent applies to each multiplicand within the product. Symbolically, this rule is expressed as: (ab) = a x b.
Example:
The Power of a Quotient Rule
This exponent rule applies to expressions written as fractions or quotients raised to a power. It states that both the numerator and the denominator are raised to that power individually.
Symbolically, the power of a quotient law is represented as: (a/b) = a/ b.
Example:
While these are some of the most frequently taught exponent rules, there are others you might encounter, like the fractional exponent law, which incorporates additional mathematical symbols such as radicals.
