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 21

Converting a root of a root into a single root

```\sqrt[m]{\sqrt[n]{a}} = \sqrt[nm]{a}```

Once again, by working backwards from the value of these two expressions we can see why they are equal. If ``\sqrt[m]{\sqrt[n]{a}} = x``, then we can construct ``a`` out of combinations of ``x`` and see how the whole equation works. To make things simple, we'll start with given values of ``m`` and ``n``. If ``\sqrt[2]{\sqrt[3]{a}} = x``, then ``x = \sqrt[2]{x*x}``, which also means that ``\sqrt[2]{\sqrt[3]{(x*x)*(x*x)*(x*x)}} = x``. So ``a = (x*x)*(x*x)*(x*x) = x^6 = x^{mn}``. And happily, ``\sqrt[mn]{x^{mn}} = x`` by definition, so we have ``\sqrt[m]{\sqrt[n]{a}} = x = \sqrt[mn]{x^{mn}}``. Our example is a specific case where ``m = 2`` and ``n = 3``, but since ``a`` will always be equal to ``x^{mn}``, the equation holds regardless of the values of ``m`` and ``n``

```\sqrt[2]{\sqrt[3]{729}} = \sqrt[2]{9} = 3 = \sqrt[6]{729}```
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A little bit about

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.

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


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