To solve the infinite nested cube root expression (\sqrt[3]{x + 3\sqrt[3]{x + 3\sqrt[3]{x + \dotsb}}}), follow these steps:

Step 1: Define the expression as (y)

Let (y = \sqrt[3]{x + 3\sqrt[3]{x + 3\sqrt[3]{x + \dotsb}}}).

Since the expression is infinite, the inner nested part is identical to (y). Thus:
(y = \sqrt[3]{x + 3y})

Step 2: Cube both sides

Eliminate the cube root by cubing both sides:
(y^3 = x + 3y)

Rearranged: (y^3 - 3y - x = 0)

Step 3: Assume a common value for (x)

A typical problem uses (x=2) (a standard case). Substitute (x=2):
(y^3 - 3y - 2 = 0)

Factor the equation:
((y+1)^2(y-2) = 0)

The real root is (y=2) (since ((y+1)^2 \geq 0)).

Answer: (\boxed{2})

作者声明:本文包含人工智能生成内容。