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Rule

Roots of Powers

To simplify an root, the radicand must first be expressed as a power. If the index of the radical and the power of the radicand are equal such that the radical expression can be simplified as follows.

The absolute value of a number is always non-negative, so when is even, the result will always be non-negative. Consider a few example roots that can be simplified by using the formula.

Is even? Simplify

Proof

Informal Justification

To write the expression for there are two cases to consider.

Both cases will be considered one at a time.

Start by noting that since is non-negative, is also non-negative. This means that the root can be rewritten using a rational exponent.
Since both and are rational numbers, the Power of a Power Property for Rational Exponents can be applied to simplify the obtained expression.
Simplify

Therefore, is equal to if is non-negative.

In case of a negative value of there are also two cases two consider.

  • is even
  • is odd

Even

Recall that a root with an even index is defined only for non-negative numbers. Although is negative, is positive. Also, a power with a negative base and an even exponent can be rewritten as a power with a positive base.
Now, since is positive, the Power of a Power Property for Rational Exponents can be applied again to simplify
Simplify

Odd

A root with an odd index is defined for all real numbers. By the definition of the root, the expression is the number that, when multiplied by itself times, will result in
Because is odd and is negative, is also negative. This means that the best candidate for is simply

Summary

If is non-negative, is always equal to However, in case of negative the value of depends on the parity of

Even
Odd

To conclude, for odd values of the expression is equal to On the other hand, if is even, can be written as

Extra

Simplifying

Depending on the index of the root and the power in the radicand, simplifying may be problematic. Because real roots with an even index are defined only for non-negative numbers, the absolute value is sometimes needed.

If is even, is defined only for non-negative

The following applet presents a decision tree to simplify In this applet, is the greatest common factor of and
Simplifying nth root of a to the mth power
Consider example roots that can be simplified by using the decision tree.
Simplify