Find the smallest perfect square that is divisible by 4,5 and 8 also find jts square roots

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.

Learn how to find the smallest perfect square that is divisible by 4, 5, and 8. This problem involves determining the least common multiple (LCM), understanding perfect squares, and calculating square roots. Suitable for high school students and anyone interested in number theory and perfect squares.

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