Question

Solve each equation and check of extr 2(7x-1)^((1)/(3))-4=0

Answer 1

To solve the equation, we isolate the cube root term and perform algebraic operations including cubing both sides to eliminate the cube root. After finding the solution x = 9/7, we substitute back into the original equation to verify that the solution is correct and reasonable, and it is not an extraneous solution.

Step-by-step explanation:

To solve the equation 2(7x - 1)1/3 - 4 = 0 and check for extraneous solutions, we first isolate the cube root term:

1. Add 4 to both sides to obtain: 2(7x - 1)1/3 = 4
2. Then divide both sides by 2 to get: (7x - 1)1/3 = 2
3. To eliminate the cube root, cube both sides of the equation to find: 7x - 1 = 23
4. Simplify this to 7x - 1 = 8
5. Add 1 to both sides to get 7x = 9
6. Finally, divide both sides by 7 to solve for x: x = 9/7

To check if this solution is reasonable, substitute x back into the original equation:

1. 2(7(9/7) - 1)1/3 - 4 = 0 simplifies to 2(2)1/3 - 4 = 0
2. Calculating 21/3 gives us 1.2599 approximately, so 2 * 1.2599 - 4 is very close to 0.

Our solution x = 9/7 is therefore correct and does not appear to be an extraneous solution.