# Algebrarules.com

The most useful rules of basic algebra,
free, simple, & intuitively organized
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Howdy! Here are a few very handy rules of algebra. These basic rules are useful for everything from figuring out your gas mileage to acing your next math test — or even solving equations from the far reaches of theoretical physics. Happy calculating!

## Algebra Rule 23

#### The nth root of the nth power of a number equals that number, or its absolute value

$$\sqrt[n]{a^n} = a,$$if n is odd;$$\sqrt[n]{a^n} = |a|,$$if n is even
Description:

If $a$ is a positive number, then $\sqrt[n]{a^n}$ will always equal $a$. However, if $a$ is negative, then the result will be positive if $n$ is even, but it will be negative if $n$ is odd. This comes from the fact that multiplying a negative number an even number of times always produces a positive result; an odd number of multiplications will produce a negative result. (This has the interesting side effect that there are no real numbers that are even-numbered roots of a negative number.)

Positive number, even root/exponent:$$\sqrt[2]{3^2} = \sqrt{9} = 3$$Positive number, odd root/exponent:$$\sqrt[3]{3^3} = \sqrt[3]{27} = 3$$Negative number, even root/exponent:$$\sqrt[2]{-3^2} = \sqrt[2]{-3*-3} = \sqrt[2]{9} = 3$$Negative number, odd root/exponent:$$\sqrt[3]{-3^3} = \sqrt[3]{-3*-3*-3} = \sqrt[3]{-27} = -3$$
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## A little bit about algebrarules.com

Algebra rules is a project by two of the folks who run The Autodidacts.

A couple of autodidact math enthusiasts, we were looking for all the rules of basic algebra concisely presented in one place. We couldn’t find such a place, so we made Algebrarules.com

These simple rules — applied with a pinch of imagination and a dash of arithmetic — can divide, conquer, and solve just about any practical algebra problem.

If you find errata in the math, bugs in the code of Algebrarules.com, or just want to say Eh, please send us a letter or join us on our roost: @rulesofalgebra.