To solve the problem, we need to find the smallest perfect square that is divisible by 4, 5, and 8, and then find its square roots.

Step 1: Find the Least Common Multiple (LCM)

First, find the LCM of 4, 5, and 8.

• $4 = 2^2$
• $5 = 5^1$
• $8 = 2^3$

The LCM of 4, 5, and 8 is: $\text{LCM}(4, 5, 8) = 2^3 \times 5^1 = 40$

Step 2: Make the LCM a Perfect Square

The next step is to make 40 a perfect square. Currently, the prime factorization of 40 is $2^3 \times 5^1$.

To make it a perfect square, we need to have even powers for all prime factors:

• The power of 2 in 40 is 3, so we need one more factor of 2 to make it $2^4$.
• The power of 5 in 40 is 1, so we need one more factor of 5 to make it $5^2$.

Thus, we multiply 40 by $2 \times 5 = 10$ to get: $40 \times 10 = 400 = (2^4) \times (5^2)$

Step 3: Square Roots

The square root of 400 is: $\sqrt{400} = 20$

Final Answer

The smallest perfect square divisible by 4, 5, and 8 is 400, and its square roots are ±20.

Would you like more details or have any questions? Here are 5 related questions to consider:

1. How do you find the prime factorization of a number?
2. What is the relationship between LCM and perfect squares?
3. Why do we need even powers for all prime factors to form a perfect square?
4. How can you verify if 400 is divisible by 4, 5, and 8?
5. What are the other perfect squares divisible by 4, 5, and 8?

Tip: Always ensure that the prime factors of a perfect square have even exponents to simplify the calculation of square roots.