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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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