How do you simplify  root5(-1988)?

Aug 13, 2016

$\sqrt[5]{- 1988} = - \sqrt[5]{1988}$ cannot be simplified further.

Explanation:

Factoring $1988$ into prime factors we find:

$1988 = 2 \cdot 2 \cdot 7 \cdot 71$

Since this has no factors with $5$th powers, there is no positive factor we can "move outside" the radical.

However, we do know ${\left(- 1\right)}^{5} = - 1$, so it is possible to move the minus sign (i.e. factor $- 1$) outside of the radical.

Spelled out we have:

$\sqrt[5]{- 1988} = \sqrt[5]{{\left(- 1\right)}^{5} \cdot 1988} = \sqrt[5]{- 1} \cdot \sqrt[5]{1988} = - 1 \cdot \sqrt[5]{1988} = - \sqrt[5]{1988}$