# Rules of exponents

Categories: exponentials

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

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