The problem involves a right-angled triangle, so we use the Pythagorean theorem. The theorem states that the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides ($

The problem involves a right-angled triangle, so we use the Pythagorean theorem. The theorem states that the square of the hypotenuse ($c$) is equal to the sum of the squares of the other two sides ($a$ and $b$):

$$c^2 = a^2 + b^2$$

From the diagram, the legs are $AC = 3\text{ cm}$ and $BC = 4\text{ m}$. The hypotenuse is $AB = x$.
To use the Pythagorean theorem, all lengths must be in the same unit. Since the hypotenuse $x$ is labeled in 'cm', we will convert $BC$ to centimeters.

$$BC = 4\text{ m} = 4 \times 100\text{ cm} = 400\text{ cm}$$

So, $a = AC = 3\text{ cm}$, $b = BC = 400\text{ cm}$, and $c = AB = x\text{ cm}$.

$$x = \sqrt{160009\text{ cm}^2}$$ $$x \approx 400.0112\text{ cm}$$

Rounding to two decimal places:

$$x \approx 400.01\text{ cm}$$

The value of side $AB$ is approximately $\boxed{\text{400.01 cm}}$.