1. \( 2(x+1)^{3 / 2}=\frac{27}{4} \)

To solve the equation \( 2(x+1)^{3/2} = \frac{27}{4} \), let's first isolate the term with \( x \). Divide both sides by 2: \[ (x+1)^{3/2} = \frac{27}{8} \] Next, we can eliminate the exponent by raising both sides to the power of \( \frac{2}{3} \): \[ x + 1 = \left(\frac{27}{8}\right)^{\frac{2}{3}} \] Calculating \( \left(\frac{27}{8}\right)^{\frac{2}{3}} \): \[ \frac{27^{2/3}}{8^{2/3}} = \frac{(3^3)^{2/3}}{(2^3)^{2/3}} = \frac{3^2}{2^2} = \frac{9}{4} \] Thus: \[ x + 1 = \frac{9}{4} \] Subtracting 1 from both sides: \[ x = \frac{9}{4} - 1 = \frac{9}{4} - \frac{4}{4} = \frac{5}{4} \] Therefore, the solution is: \[ \boxed{\frac{5}{4}} \]