Reducing to the same index

To compare or multiply radicals with different indices, they are reduced to the same index (the lowest common multiple of the indices), passing through fractional exponents:

$\sqrt{a}\cdot\sqrt[3]{b} = a^{\frac{1}{2}}\cdot b^{\frac{1}{3}} = a^{\frac{3}{6}}\cdot b^{\frac{2}{6}} = \sqrt[6]{a^3\cdot b^2}.$

Example

$\sqrt{x-2}\cdot\sqrt[3]{(x+1)^2}$, with domain of existence $x\ge 2$. $(x-2)^{1/2}\cdot(x+1)^{2/3} = (x-2)^{3/6}\cdot(x+1)^{4/6} = \sqrt[6]{(x-2)^3(x+1)^4}$

Remark — Sign function

When a radical of odd index is brought under an even index, one must multiply by the sign function: $\operatorname{Segno}(f) = \begin{cases} +1 & \text{se } f\ge 0 \\ -1 & \text{se } f<0 \end{cases}$ Example: $\sqrt[3]{1-x} = \sqrt[6]{(1-x)^2}\cdot\operatorname{Segno}(1-x)$.

Links

Topics: Radicals
Concepts: Sign function · Root index · Radical
Skills: Simplifying