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

To solve the equation: \[ 2(x + 1)^{\frac{3}{2}} = \frac{27}{4} \] follow these steps: ### Step 1: Isolate the Term with the Exponent Divide both sides of the equation by 2 to isolate \((x + 1)^{\frac{3}{2}}\): \[ (x + 1)^{\frac{3}{2}} = \frac{27}{8} \] ### Step 2: Eliminate the Fractional Exponent To remove the exponent \(\frac{3}{2}\), raise both sides of the equation to the power of \(\frac{2}{3}\): \[ \left( (x + 1)^{\frac{3}{2}} \right)^{\frac{2}{3}} = \left( \frac{27}{8} \right)^{\frac{2}{3}} \] Simplifying the left side: \[ x + 1 = \left( \frac{27}{8} \right)^{\frac{2}{3}} \] ### Step 3: Simplify the Right Side Recognize that \(27 = 3^3\) and \(8 = 2^3\): \[ \left( \frac{27}{8} \right)^{\frac{2}{3}} = \left( \frac{3^3}{2^3} \right)^{\frac{2}{3}} = \frac{3^{3 \times \frac{2}{3}}}{2^{3 \times \frac{2}{3}}} = \frac{3^2}{2^2} = \frac{9}{4} \] So, \[ x + 1 = \frac{9}{4} \] ### Step 4: Solve for \(x\) Subtract 1 from both sides: \[ x = \frac{9}{4} - 1 = \frac{9}{4} - \frac{4}{4} = \frac{5}{4} \] ### Final Answer \[ x = \frac{5}{4} \]