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Why are exponential-time problems called intractable?

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A three-dimensional Lorenz attractor graph with swirling blue lines forming a butterfly-shaped pattern, representing chaotic motion in a mathematical weather model, with labeled x, y, and z axes. — complexity The weather model of meteorologist Edward Lorenz
The x-, y-, and z-axes correspond to components of the weather, while the graph indicates feasible weather patterns. The graph is a function of time, which means, in theory, that tracing the curve in one direction reveals past weather patterns, while tracing the curve in the other direction predicts future weather patterns. However, practically indistinguishable points may lead to completely different weather epochs, indicated by the two distinct “lobes.” This phenomenon, known as sensitivity to initial conditions, is what prevents precise weather forecasts more than a few days into the future.

Exponential-time problems are classified as intractable because they are impractical for computers to solve as the size of the input increases. While problems solvable in polynomial time are considered efficient, those with exponential time complexity—such as $O (2^{n})$ —experience a rapid, unsustainable growth in the number of operations required.12

Even for relatively small inputs, the computational demand of an exponential algorithm can become overwhelming. For instance, an algorithm with $O (2^{n})$ complexity might execute in less than a second for an input size of $n = 10$ , but it could take more than 40,000 years to complete if the input size increases to $n = 100$ , assuming a computer capable of performing $10^{18}$ operations per second. Because these solutions cannot be implemented efficiently, computer scientists often turn to heuristic algorithms to find near-optimal solutions within a reasonable timeframe.12