Invariant Property of Radicals
The value of a radical remains unchanged when both the index of the root and the exponent of the radicand are multiplied or divided by the same nonzero natural number p. $$ \sqrt[n]{a^m} = \sqrt[n \cdot p]{a^{m \cdot p }} = \sqrt[ \frac{n}{p} ]{a^{ \frac{m}{p} }} $$
A radical obtained through this transformation is called an equivalent radical, because the expressions represent the same numerical value.
A Practical Example
Consider the cube root of 8.
$$ \sqrt[3]{8} = 2 $$
Since 8 = 23, rewrite the radicand in exponential form.
$$ \sqrt[3]{2^3} = 2 $$
Now multiply both the index of the root and the exponent of the radicand by p = 2.
$$ \sqrt[3 \cdot 2]{2^{3 \cdot 2}} $$
$$ \sqrt[6]{2^{6}} $$
The transformed radical is equivalent, as its value remains unchanged.
$$ \sqrt[6]{2^{6}} = 2 $$
Note. Dividing both the index and the exponent by 2 restores the original radical. $$ \sqrt[\frac{6}{2}]{2^{\frac{6}{2}}} = 2 $$ $$ \sqrt[3]{2^{3}} = 2 $$
Proof
Assume the equality between the following radicals.
$$ \sqrt[n]{a^m} = \sqrt[n \cdot p]{a^{m \cdot p}} $$
Raise both sides of the equation to the power n·p.
$$ ( \sqrt[n]{a^m} )^{n \cdot p} = ( \sqrt[n \cdot p]{a^{m \cdot p}} )^{n \cdot p} $$
On the right-hand side, the radical simplifies directly.
$$ ( \sqrt[n]{a^m} )^{n \cdot p} = a^{m \cdot p} $$
Apply the exponent rule \( (x^n)^p = x^{n \cdot p} \) to the left-hand side.
$$ [ ( \sqrt[n]{a^m} )^n ]^p = a^{m \cdot p} $$
Simplify the radical expression.
$$ [ a^m ]^p = a^{m \cdot p} $$
Apply the power rule once more.
$$ a^{m \cdot p} = a^{m \cdot p} $$
The equality is verified.
Therefore, the original radicals are equivalent radicals.
$$ \sqrt[n]{a^m} = \sqrt[n \cdot p]{a^{m \cdot p}} $$
Radicals with a Negative Radicand
The invariant property cannot be applied without caution when the radicand is negative.
Example
Radicals with an odd index admit negative radicands.
$$ \sqrt[3]{-8} = -2 \quad \Longrightarrow \quad (-2)^3 = (-2)\cdot(-2)\cdot(-2) = -8 $$
Now apply the invariant property by multiplying both the index and the exponent by 2.
$$ \sqrt[3 \cdot 2]{(-8)^{1 \cdot 2}} $$
$$ \sqrt[6]{(-8)^2} $$
$$ \sqrt[6]{64} = 2 $$
After the transformation, the result is 2. However, 23 = 8, which is not -8.
This inconsistency shows that the invariant property cannot be directly applied to negative radicands.
Note. If and only if the index n is odd, the radicand may be rewritten as a product of -1 and a positive quantity. $$ \sqrt[3]{-8} = \sqrt[3]{-1 \cdot 8} $$ Since \( (-1)^3 = -1 \), the factor -1 can be extracted from the radical. $$ \sqrt[3]{-8} = \sqrt[3]{-1 \cdot 8} = -1 \cdot \sqrt[3]{8} $$ The invariant property can now be applied safely because the remaining radicand is positive. For example, multiply the index and exponent by 2. $$ \sqrt[3]{-8} = -1 \cdot \sqrt[3]{8} = -1 \cdot \sqrt[3 \cdot 2]{8^2} = - \sqrt[6]{64} = -2 $$ The result is correct.
And so on.
