square root of 7500

Steps to Solve

1. Factor the number 7500

To find the square root, we should factor 7500 into its prime factors.
We can divide 7500 by 100 first:
$$ 7500 \div 100 = 75 $$
Next, we can factor 75:
$$ 75 = 3 \times 5^2 $$
Now combine these factors:
$$ 7500 = 75 \times 100 = 3 \times 5^2 \times (10^2) = 3 \times 5^2 \times (2 \times 5)^2 $$
Thus,
$$ 7500 = 3 \times 2^2 \times 5^4 $$

2. Use the prime factorization to find the square root

The square root of a number can be found by taking the square root of each factor.
From our factorization, we have:
$$ \sqrt{7500} = \sqrt{3 \times 2^2 \times 5^4} $$

Breaking it down: $$ \sqrt{7500} = \sqrt{3} \times \sqrt{2^2} \times \sqrt{5^4} $$

3. Calculate the square roots of the factors

Now, compute the square roots:

• The square root of $2^2$ is $2$.
• The square root of $5^4$ is $25$.

So we have:
$$ \sqrt{7500} = \sqrt{3} \times 2 \times 25 $$

4. Multiply the factors to get the final answer

Combine the results:
$$ \sqrt{7500} = 50\sqrt{3} $$