Taking factors outside the radical sign

A radical is simplified by taking outside the radical sign those factors whose exponent is a multiple of (or exceeds) the index.

Property — Taking factors outside

To take factors outside the radical, perform the integer division of the exponent by the index: $\sqrt[n]{a^k} = a^q\cdot\sqrt[n]{a^r}, \qquad k = nq+r,\; 0\le r < n.$

Example

$\sqrt[4]{2^6} = 2^{6/4} = 2^{1+2/4} = 2\cdot\sqrt[4]{2^2} = 2\sqrt[4]{4}$ because $6\div 4 = 1$ with remainder $2$.

$\sqrt[7]{3^{15}\cdot 5^{12}} = 3^2\cdot 5^1\cdot\sqrt[7]{3^1\cdot 5^5} = 45\sqrt[7]{3\cdot 5^5}$ because $15\div 7 = 2$ remainder $1$ and $12\div 7 = 1$ remainder $5$.

Caution — With literal expressions and an even index

If the index is even and the base may be negative, the absolute value is needed: $\sqrt[4]{(x-2)^6} = |x-2|\cdot\sqrt[4]{(x-2)^2}$

Links

Topics: Radicals
Concepts: Radical · Absolute value
Skills: Simplifying