We have: #root(4)(x^(2) + 6 x) - 2 = 0#
#Rightarrow (x^(2) + 6 x)^(frac(1)(4)) - 2 = 0#
First, let's add #2# to both sides of the equation:
#Rightarrow (x^(2) + 6 x)^(frac(1)(4)) = 2#
Then, let's raise both sides to #4#:
#Rightarrow ((x^(2) + 6 x)^(frac(1)(4)))^(4) = (2)^(4)#
#Rightarrow x^(2) + 6 x = 16#
Now, let's subtract both sides by #16#, and then use the quadratic formula:
#Rightarrow x^(2) + 6 x - 16 = 0#
#Rightarrow x = frac(- 6 pm sqrt((6)^(2) - 4(1)(- 16))(2(1))#
#Rightarrow x = frac(- 6 pm sqrt(36 + 64))(2)#
#Rightarrow x = frac(- 6 pm sqrt(100))(2)#
#Rightarrow x = frac(- 6 pm 10)(2)#
#Rightarrow x = - 8, 2#
Finally, let's verify these solutions by substituting them back into the original equation:
#Rightarrow root(4)((- 8)^(2) + 6 cdot (- 8)) - 2 = 0#
#Rightarrow root(4)(64 - 48) - 2 = 0#
#Rightarrow root(4)(16) - 2 = 0#
#Rightarrow 2 - 2 = 0#
#Rightarrow 0 = 0 " " "True"#
#and#
#Rightarrow root(4)((2)^(2) + 6 cdot (2)) - 2 = 0#
#Rightarrow root(4)(4 + 12) - 2 = 0#
#Rightarrow root(4)(16) - 2 = 0#
#Rightarrow 2 - 2 = 0#
#Rightarrow 0 = 0 " " "True"#
Therefore, the solutions to the equation are #x = - 8# and #x = 2#.
