Question 252903
{{{sqrt(49)}}} = {{{sqrt(7*7)}}} = 7.


{{{sqrt(-27)^3}}} = {{{(sqrt(3*3*-3))^3}}} = {{{(3*sqrt(-3))^3}}} = {{{3^3 * sqrt(-3)^3}}} = {{{27*sqrt(-3)^2*sqrt(-3)}}} = {{{27*-3*sqrt(-3)}}} = {{{-81*sqrt(-3)}}}


to confirm first equation:


{{{sqrt(7^2)}}} = {{{sqrt(49)}}} = 7.


to confirm second equation:


{{{sqrt((-81*sqrt(-3))^2)}}} = {{{sqrt((-81)^2*sqrt(-3)^2)}}} = {{{sqrt(6561*(-3))}}} = {{{sqrt(-19683)}}}.


{{{sqrt(-19683)}}} = {{{sqrt((-27)^3)}}}


{{{sqrt((-27)^3)}}} is the same as {{{(sqrt(-27))^3}}}.


this can be seen easier with exponential notation.



{{{sqrt((-27)^3)}}} is the same as {{{((-27)^3)^(1/2)}}} which equals {{{(-27)^(3*.5)}}} which equals {{{(-27)^(1.5)}}}.



{{{sqrt(-27)^3}}} is the same as {{{((-27)^(1/2))^3)}}} which equals {{{(-27)^(.5*3)}}} which equals {{{(-27)^(1.5)}}}.