Question 317101
It's not much of a simplification (in my opinion) as the expression doesn't get any simpler, but here it goes...



{{{ c/sqrt(c^2 - v^2) }}} Start with the given expression.



{{{ (c/c)/(sqrt(c^2 - v^2)/c) }}} Divide both the numerator and the denominator by 'c'



{{{ 1/(sqrt(c^2 - v^2)/c) }}} Reduce {{{c/c}}} to get 1.



{{{ 1/(sqrt(c^2 - v^2)/sqrt(c^2)) }}} Rewrite {{{c}}} as {{{sqrt(c^2)}}}. Note: this implies that 'c' is a non-negative number, which it is.



{{{ 1/(sqrt((c^2 - v^2)/(c^2))) }}} Combine the lower square roots using the identity {{{sqrt(x)/sqrt(y)=sqrt(x/y)}}}



{{{ 1/(sqrt((c^2)/(c^2) - (v^2)/(c^2))) }}} Break up the lower inner fraction.



{{{ 1/(sqrt(1 - (v^2)/(c^2))) }}} Reduce {{{(c^2)/(c^2)}}} to get 1.



So {{{ c/sqrt(c^2 - v^2) = 1/(sqrt(1 - (v^2)/(c^2))) }}}