Solve for x in the equation 3(x + 1)^(2/3) = 12.

To solve the equation ( 3(x + 1)^{2/3} = 12 ), we first isolate the term containing ( x ). We achieve this by dividing both sides of the equation by 3:

[

(x + 1)^{2/3} = \frac{12}{3}

]

[

(x + 1)^{2/3} = 4

]

Next, to eliminate the exponent ( \frac{2}{3} ), we raise both sides to the power of ( \frac{3}{2} ):

[

\left((x + 1)^{2/3}\right)^{\frac{3}{2}} = 4^{\frac{3}{2}}

]

[

x + 1 = 4^{\frac{3}{2}}

]

Calculating ( 4^{\frac{3}{2}} ), we first find the square root of 4, which is 2, and then raise it to the third power:

[

4^{\frac{3}{2}} = (2^2)^{\frac{3}{2}} = 2^3 = 8