Substitute \(u = 3^x\) into the equation:

\(u + u^2 = u^3\).

Rearrange to form a quadratic equation:

\(u^3 - u^2 - u = 0\).

Factor out \(u\):

\(u(u^2 - u - 1) = 0\).

Since \(u = 3^x\) cannot be zero, solve \(u^2 - u - 1 = 0\).

Using the quadratic formula \(u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1, b = -1, c = -1\):

\(u = \frac{1 \pm \sqrt{1 + 4}}{2} = \frac{1 \pm \sqrt{5}}{2}\).

Select the positive root: \(u = \frac{1 + \sqrt{5}}{2}\).

Convert back to \(x\) using \(u = 3^x\):

\(3^x = \frac{1 + \sqrt{5}}{2}\).

Take the logarithm of both sides:

\(x \log 3 = \log \left( \frac{1 + \sqrt{5}}{2} \right)\).

Solve for \(x\):

\(x = \frac{\log \left( \frac{1 + \sqrt{5}}{2} \right)}{\log 3}\).

Calculate \(x\) to 3 significant figures:

\(x \approx 0.438\).