Solve by factoring

Question

(3z-4)^(-1/3) -5(3z-4)^(-4/3)=0 solve by factoring?

Answer 1

(3z-4)^(-1/3) - 5(3z-4)^(-4/3)=0

(3z-4)^(-1/3) - 5((3z-4)^(-1/3))^4=0

 

Factor out Greatest Common Factor of (3z-4)^(-1/3):

((3z-4)^(-1/3)) * ( 1 - 5((3z-4)^(-1/3))^3 ) = 0

 

Simplifying a little:

((3z-4)^(-1/3)) * ( 1 - 5/(3z-4)) = 0

 

Using Zero-Product Rule:

Case 1: (3z-4)^(-1/3) = 0 => 3z = 4 => z = 4/3

Case 2: 1 - 5/(3z-4) = 0 => 3z - 4 = 5 => z = 3

 

There are two answers: z = 4/3 and z = 3.

 

Check: 

Case 1: (3(4/3)-4)^(-1/3) - 5(3(4/3)-4)^(-4/3) =? 0

               (4 - 4)^(-1/3) - 5(4-4)^(-4/3) = 0 - 0 = 0 √

Case 2: (3(3)-4)^(-1/3) - 5(3(3)-4)^(-4/3) =? 0
                    5^(-1/3) - 5(5)^(-4/3) = 5^(-1/3) - 5^(3/3 - 4/3) = 5^(-1/3) - 5^(-1/3) = 0 √

Answer 2

Hi Jette,

 

Sorry I made a mistake in my earlier answer.  If you divide by (3z-4)^(-1/3) like I had suggested you lose one of the two answers for z.  

 

Steve has written out a thorough step by step solution to get the correct answer.