The most useful rules of basic algebra,
free, simple, & intuitively organized

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

```\tiny\text{if n is odd:} \newline \normalsize \sqrt[n]{a^n} = a \newline \tiny\text{if n is even:} \newline \normalsize \sqrt[n]{a^n} = |a|```

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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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

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.

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