(x-9)^1/4-(x+2)^1/3=-1 Find x !

For the beggining, we'll impose constraint of existence of
radical (x-9)^1/4.

x - 9 >=
0

x >= 9

So, the
solutions of the equation have to be located in the range [9 ;
+infinite).

NOw, to solve the equation, we'll note a =
(x-9)^1/4 and b = (x+2)^1/3.

We'll raise a to the 4th
power:

a^4 = x - 9 (1)

We'll
raise b to the 3rd power:

b^3 = x + 2
(2)

We'll subtract (2) from (1) and we'll
get:

a^4 - b^3 = x - 9 - x -
2

a^4 - b^3 = -11 (3)

a - b =
-1 => b = a + 1

We'll substitute b in
(3):

a^4 - (a+1)^3 = -11

a^4 -
a^3 - 3a^2 - 3a - 1 = -11

a^4 - a^3 - 3a^2 - 3a + 10 =
0

We'll factorize and we'll
get:

(a-2)(a^3 +a^2 - a - 5) =
0

a^3 +a^2 - a - 5 is
irreducible

a - 2 = 0

a =
2

(x-9)^1/4 = 2

x - 9 =
16

x = 16+9

x =
2

25 is located in the range [9 ;
infinite).

We'll accept it as
solution of the equation, so x = 25.