We are given the expression with radical sign.
#color(blue)(root(9)(27a^3b^6))# #..color(red)(Expression.1)#
We will be using the following formulas to simplify:
#color(green)(root(n)(a) = a^(1/n))# #..color(blue)(.. 1)#
#color(green)((a^m)^n = a^(mn))# #..color(blue)(.. 2)#
#color(green)((a^m)^(1/n) = a^(m/n))# #..color(blue)(.. 3)#
#color(green)((a*b*c)^m = a^m * b^m * c^m# #..color(blue)(.. 4)#
We write the #..color(red)(Expression.1)# as follows:
#color(blue)((27a^3b^6)^(1/9))# ..... using formula #..color(blue)(.. 1)#
In the next step, we will isolate each factor inside the parenthesis with the exponent as follows:
#color(blue)((27)^(1/9)* (a^3)^(1/9)* (b^6)^(1/9)# ..... using formula #..color(blue)(.. 4)#
We can now rewrite the above expression as follows:
#color(blue)((3^3)^(1/9)* (a^3)^(1/9)* (b^6)^(1/9)#
#color(blue)(rArr 3^(3/9) * a^(3/9)*b^(6/9))# ..... using formula #..color(blue)(.. 3)#
After simplification, we can write the above expression as:
#color(blue)(rArr 3^(1/3) * a^(1/3)*b^(2/3))#
