What is the square root of 18000?

Steps to Solve

1. Factor the number 18000

To find the square root of 18000, we can start by factoring it into its prime components.

$$ 18000 = 18 \times 1000 = 18 \times 10^3 $$

Next, we can break down 18 and 1000:

$$ 18 = 2 \times 3^2 $$

$$ 1000 = 10^3 = (2 \times 5)^3 = 2^3 \times 5^3 $$

So, we have:

$$ 18000 = 2^1 \times 3^2 \times 2^3 \times 5^3 $$

2. Combine the prime factors

Now we can combine the factors:

$$ 18000 = 2^{1+3} \times 3^2 \times 5^3 = 2^4 \times 3^2 \times 5^3 $$

3. Calculate the square root

To find the square root, we take half of the exponents of the prime factors:

$$ \sqrt{18000} = \sqrt{2^4} \times \sqrt{3^2} \times \sqrt{5^3} $$

Calculating each term gives:

$$ \sqrt{2^4} = 2^2 = 4, \quad \sqrt{3^2} = 3, \quad \text{and} \quad \sqrt{5^3} = 5^{3/2} = 5 \sqrt{5} $$

So we can combine these together:

$$ \sqrt{18000} = 4 \times 3 \times 5\sqrt{5} $$

4. Final multiplication and simplification

Now we multiply the numbers outside the square root:

$$ 4 \times 3 \times 5 = 60 $$

Therefore, our final answer can be expressed as:

$$ \sqrt{18000} = 60\sqrt{5} $$