To solve this problem, we need to do something with that 2/3 power first.
Since the 2/3 encompasses an expression using addition, the power does not distribute. So we have to do a little manipulation:
(x+9)^(2/3)=4 => root(3)((x+9)^(2))=4
Now that we have that, we can start to simplify this a little.
Since the cube root encompasses the entirety of the left side, we can cube the whole thing to get rid of it.
color(blue)(Note:"What you do to one side, you must do to the other".
We should now have this:
root(3)((x+9)^(2))=4
=>(root(3)((x+9)^(2)))^3=4^3
=>(x+9)^2=64
From here we need to put the (x+9)^2 into standard form, and solve from there:
(x+9)^2=64
=>x^2+18x+81=64
=>x^2+18x+15=0
To solve this, we can use the quadratic formula, which is defined as:
-color(red)(b)+-sqrt(color(red)(b)^2-4color(blue)(a)color(green)(c))/(2color(blue)(a))
color(blue)(a=1) color(red)(b=18) color(green)(c=15)
=(-color(red)(18)+-sqrt(color(red)(18)^2-4(color(blue)(1))(color(green)(15))))/(2(color(blue)(1))
=-(18+-sqrt(324-60))/(2)
=(-18+-2sqrt(66))/2
=-9+-sqrt(66)
Hope this helped!
