1. **Problem statement:** Find the dimensions of the smallest square that can be tiled using 18 cm by 24 cm tiles without cutting. 2. **Understanding the problem:** We want the smallest square whose side length is a multiple of both 18 and 24 cm. 3. **Formula used:** The side length of the smallest square is the least common multiple (LCM) of 18 and 24. 4. **Calculate prime factorizations:** $$18 = 2 \times 3^2$$ $$24 = 2^3 \times 3$$ 5. **Find the LCM:** Take the highest powers of each prime: $$\text{LCM} = 2^3 \times 3^2 = 8 \times 9 = 72$$ 6. **Interpretation:** The smallest square has side length 72 cm. 7. **Answer:** The smallest square that can be tiled with 18 cm by 24 cm tiles has dimensions: $$\boxed{72 \text{ cm} \times 72 \text{ cm}}$$