Rules of exponents

This article looks at exponents and the exponential function.

Here is a video:

What are exponential functions

An exponential function in x takes the form a raised to the power x, that is $a^x$. a is called the base, and x is called the exponent.

Here are the first few powers of a:

$a^2$, of course, is commonly called a squared and is calculated as a multiplied by a.

$a^3$ is commonly called a cubed and is calculated as a multiplied by a multiplied by a.

$a^4$ doesn't have a special name, it can be called the fourth power of a, or a to the power four.

Simple definition of the exponential function

Generally, $a^n$, where n is an integer of 2 or greater is the nth power of a and is often described as a multiplied by itself n times.

That isn't quite true though. a squared is actually a multiplied by itself once, not twice.

A better definition would be that $a^n$ is one multiplied by a, n times.

Exponential values

If we chose a base of 2, we can calculate some values:

As another example, if we chose a base of 10, we can get different values:

Laws of exponents

There are several laws that can be used to simplify expressions involving exponents:

• Product of powers rule.
• Quotient of powers rule.
• Power of a power rule.
• Power of a product rule.
• Power of a quotient rule.

Product of powers rule

When multiplying two exponential terms that have the same base, we add the exponents:

$$ a^p a^q = a^{p + q} $$

Here is an illustration for $a^3$ multiplied by $a^2$:

Expanding each term shows that the total number of terms is 5, the sum of the exponents.

Quotient of powers rule

When dividing two exponential terms that have the same base, subtract the exponent of the denominator from the exponent of the numerator:

$$ \frac{a^p}{a^q} = a^{p - q} $$

Here is an illustration for $a^5$ divided by $a^3$:

Expanding each term and cancelling top and bottom shows that the total number of terms is 2, the difference between the exponents.

Power of a power rule

When raising a power to a power, multiply the two exponents:

$$ {a^p}^q = a^{p q} $$

Here is an illustration for $a^3$ squared:

Expanding each term and cancelling top and bottom shows that the total number of terms is 6, the product of the exponents.

Power of a product rule

If we have a base that consists of a product of two (or more) terms, all raised to a power, we can simplify it by distributing the exponent to each term in the base:

$$ (a b)^p = a^p b^p $$

Here is an example of ab cubed:

Expanding and rearranging the equation for ab cubed demonstrates the rule. This can easily be extended to other powers.

Power of a quotient rule

The previous rule also applies to the quotient of two values, all raised to a power. Again we can simplify it by distributing the power to each term in the quotient:

$$ (\frac{a}{b})^p = \frac{a^p}{b^p} $$

Here is an example of a/b cubed:

Expanding and rearranging the equation for a/b cubed demonstrates the rule. This can easily be extended to other powers.

See also

• Negative and fractional exponents
• Graphs of exponential functions
• 2 to the power π - exponential function for an irrational value of x
• What is e?
• Formula for e
• Derivative of the exponential function
• Proof that e is irrational