Start by writing your expression
#sqrt(7) + sqrt(7""^2) + sqrt(7""^3) + sqrt(7""^4) + sqrt(7""^5)#
Now, one way in which you can simplify this expression is by using the product property of radicals, which tells you that
#sqrt( a * b) = sqrt(a) * sqrt(b)#
for #a>=0# and #b>=0#.
This means that you can write all the numbers that are under the radicals as products of #7""^2#, starting with
#color(blue)(7""^3 = 7""^2 * 7)#
#color(green)(7""^4 = 7""^2 * 7""^2)#
#color(orange)(7""^5 = 7""^2 * 7""^2 * 7)#
Your expression will become
#sqrt(7) + 7 + color(blue)(sqrt(7""^2 * 7)) + color(green)(sqrt(7""^2 * 7""^2)) + color(orange)(sqrt(7""^2 * 7""^2 * 7))#
#sqrt(7) + 7 + color(blue)(sqrt(7""^2) * sqrt(7)) + color(green)(sqrt(7""^2) * sqrt(7""^2)) + color(orange)(sqrt(7""^2) * sqrt(7""^2) * sqrt(7))#
#sqrt(7) + 7 + color(blue)(7sqrt(7)) + color(green)(7 * 7) + color(orange)(7 * 7 * sqrt(7))#
#sqrt(7) + 7 + 7sqrt(7) + 49 + 49 sqrt(7)#
Now simply add the terms that do not contain a radical and the terms that do contain a radical separately to get
#(7 + 49) + (sqrt(7) + 7sqrt(7) + 49sqrt(7)) = 56 + 57sqrt(7)#
Alternatively, you can simplify this expression by grouping some of these terms together and factorizing all but the first term. This will get you
#sqrt(7) + (sqrt(7""^2) + sqrt(7""^3)) + (sqrt(7""^4) + sqrt(7""^5))#
#sqrt(7) + sqrt(7""^2) * (1 + sqrt(7)) + sqrt(7""^4) * (1 + sqrt(7))#
This is equivalent to
#sqrt(7) + (1 + sqrt(7)) * (sqrt(7""^2) + sqrt(7""^4))#
#sqrt(7) + (1 + sqrt(7)) * (7 + 49)#
#sqrt(7) + (1 + sqrt(7)) * 56#
#sqrt(7) + 56 + 56 * sqrt(7)#
Once again, the result will be
#56 + 56sqrt(7) + sqrt(7) = 56 + 57sqrt(7)#
