Bringing a Factor Inside a Radical
To bring a nonnegative factor inside a radical, multiply the exponent of the factor (m) by the index of the root (n). $$ a^m \cdot \sqrt[n]{b} = \sqrt[n]{ a^{m \cdot n} \cdot b } $$ where a≥0 and b≥0.
Negative factors cannot be brought inside the radical.
In these cases, keep the minus sign outside the radical and place inside the absolute value of the factor raised to the index of the root.
A practical example
Consider the expression
$$ a^2 \cdot \sqrt[3]{b^2} $$
To bring the factor a2 inside the radical, multiply its exponent (2) by the index of the root (3), that is 2·3=6.
$$ \sqrt[3]{ a^{2 \cdot 3} \cdot b^2} $$
$$ \sqrt[3]{ a^6 \cdot b^2} $$
Example 2
Consider the expression
$$ 2 \cdot \sqrt[3]{a^2} $$
To bring the factor 2 inside the radicand, multiply its exponent (1) by the index of the root (3).
$$ \sqrt[3]{2^{1 \cdot 3} \cdot a^2} $$
$$ \sqrt[3]{2^3 \cdot a^2} $$
$$ \sqrt[3]{8 \cdot a^2} $$
Example 3
Consider the expression
$$ -2 \cdot \sqrt[3]{a^2} $$
In this case the factor outside the radical is negative.
Keep the minus sign outside and raise the absolute value of the factor to the index of the root (3).
$$ - \sqrt[3]{2^{1 \cdot 3} \cdot a^2} $$
$$ - \sqrt[3]{2^3 \cdot a^2} $$
$$ - \sqrt[3]{8 \cdot a^2} $$
Proof
Consider the expression
$$ a^m \cdot \sqrt[n]{b} $$
The external factor am is nonnegative (a≥0) and multiplies the radical.
This factor can be viewed as the radicand of a root with index 1.
$$ \sqrt[1]{ a^m } \cdot \sqrt[n]{b} $$
Note. The first root of a number n is simply the number itself, that is n1=n. $$ \sqrt[1]{n} = n \Longleftrightarrow n^1 = n $$ In many mathematics textbooks the root with index 1 is omitted because it does not add any information. However, in this proof it helps make the reasoning clearer.
Using the invariance property of radicals, multiply both the index of the root (1) and the exponent of the radicand (m) by n.
$$ \sqrt[1 \cdot n ]{ a^{m \cdot n} } \cdot \sqrt[n]{b} $$
$$ \sqrt[ n ]{ a^{m \cdot n} } \cdot \sqrt[n]{b} $$
The two radicals now have the same index, so the product rule for radicals can be applied.
$$ \sqrt[n]{a^{m \cdot n} \cdot b} $$
And so on.
