1. **State the problem:** We need to find the length $x$ in a triangle using the sine rule. Given angles are $40^\circ$ and $56^\circ$, and the side opposite the $56^\circ$ angle is 8 cm. 2. **Find the third angle:** The sum of angles in a triangle is $180^\circ$. $$\text{Third angle} = 180^\circ - 40^\circ - 56^\circ = 84^\circ$$ 3. **Recall the sine rule formula:** $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$ where $a,b,c$ are sides opposite angles $A,B,C$ respectively. 4. **Apply the sine rule to find $x$:** Let $x$ be opposite $40^\circ$, and side 8 cm opposite $56^\circ$. $$\frac{x}{\sin 40^\circ} = \frac{8}{\sin 56^\circ}$$ 5. **Solve for $x$:** $$x = \frac{8 \times \sin 40^\circ}{\sin 56^\circ}$$ 6. **Calculate the sines:** $$\sin 40^\circ \approx 0.6428, \quad \sin 56^\circ \approx 0.8290$$ 7. **Substitute and simplify:** $$x = \frac{8 \times 0.6428}{0.8290} = \frac{5.1424}{0.8290}$$ 8. **Final calculation:** $$x \approx 6.2$$ **Answer:** The length $x$ is approximately 6.2 cm to 1 decimal place.