Even and Odd Roots of Negative Real Numbers
In the real number system, an even-index radical of a negative number is undefined. The reason is straightforward: no real number raised to an even power can produce a negative result.
For example, the expression
$$ \sqrt{-9} $$
has no real value. Indeed, squaring -3 yields a positive number rather than the radicand:
$$ (-3)^2 = (-3) \cdot (-3) = 9 $$
More generally, if the index n is even, the radical
$$ \sqrt[n]{a} $$
is defined in ℝ only when the radicand satisfies
$$ a \ge 0 $$
Remark. This limitation holds only within the real number system ℝ. In a broader number system, such as the complex numbers, even-index roots of negative real numbers are well defined.
By contrast, if the index n is odd, the radical of a negative real number is always defined in ℝ.
For example,
$$ \sqrt[3]{-27} = -3 $$
since
$$ (-3)^3 = (-3) \cdot (-3) \cdot (-3) = 9 \cdot (-3) = -27 $$
Accordingly, for odd indices,
$$ \sqrt[n]{a} \in \mathbb{R} \quad \text{for every } a \in \mathbb{R}. $$
A Particular Case
Consider the expression
$$ \sqrt[4]{(-3)^2} $$
One might be tempted to simplify the index and the exponent immediately. However, doing so would lead to
$$ \sqrt[2]{-3} $$
which is undefined in ℝ.
The key observation is that the exponent of the radicand is even. We therefore rewrite
$$ (-3)^2 = (3)^2 = |-3|^2 $$
Substituting this equivalent form gives
$$ \sqrt[4]{|-3|^2} $$
Now the index of the radical and the exponent of the radicand may be reduced:
$$ \sqrt{| -3 |} $$
Since
$$ |-3| = 3 $$
we obtain
$$ \sqrt{3} $$
In general, if n is even and m is even, then
$$ \sqrt[n]{a^m} = \sqrt[n]{|a|^m}. $$
Example
Evaluate
$$ \sqrt{(a-5)^2} $$
The expression (a-5)2 is nonnegative for every real value of a, since it is a square. Hence we rewrite it using absolute value:
$$ \sqrt{|a-5|^2} $$
Simplifying the index and the exponent yields
$$ \sqrt{|a-5|^2} = |a-5| $$
The value of the radical is therefore the absolute value |a-5|.
Note. Writing $$ \sqrt{(a-5)^2} = a-5 $$ would be incorrect. The expression a-5 may be negative, whereas an even-index radical in ℝ corresponds to the nonnegative square root. The absolute value is therefore essential.
The Invariant Property and Negative Radicands
The invariant property of radicals cannot be applied directly when the original radicand is negative.
Example
An odd-index radical admits a negative radicand:
$$ \sqrt[3]{-8} = -2 \quad \Longrightarrow \quad (-2)^3 = -8 $$
If one formally applies the invariant property by multiplying both the index and the exponent of the radicand by 2, one obtains
$$ \sqrt[6]{(-8)^2} = \sqrt[6]{64} = 2 $$
However,
$$ 2^3 = 8 \ne -8 $$
This discrepancy shows that the invariant property must not be applied when the initial radicand is negative.
A legitimate method. If and only if the index n is odd, the radicand may be factored as a product of -1 and a positive number:
$$ \sqrt[3]{-8} = \sqrt[3]{-1 \cdot 8} = \sqrt[3]{-1} \cdot \sqrt[3]{8} $$
Since
$$ (-1)^3 = -1 $$
we obtain
$$ \sqrt[3]{-8} = -1 \cdot \sqrt[3]{8} $$
In this equivalent form, the invariant property can be safely applied to the positive factor:
$$ - \sqrt[3]{8} = - \sqrt[6]{64} = -2 $$
The result is consistent with the original definition.
