Finding the Square Root of 625.0025

The question asks us to find the square root of \(625.0025\). The square root of a number is a value that, when multiplied by itself, gives the original number. We are looking for a number \(x\) such that \(x^2 = 625.0025\).

Understanding the Number and its Square Root

Let's first consider the integer part, 625. We know that \(25^2 = 25 \times 25 = 625\). So, the square root of 625 is 25.

Now, let's look at the decimal part of the number, 625.0025. It has 4 decimal places.

A useful rule for squaring and taking the square root of decimal numbers is related to the number of decimal places:

• When you square a number with \(n\) decimal places, the result has \(2n\) decimal places.
• Conversely, when you take the square root of a perfect square number with \(2n\) decimal places, the square root will have \(n\) decimal places.

In our number, \(625.0025\), there are 4 decimal places. According to the rule, its square root should have \(4/2 = 2\) decimal places. This would suggest an answer like 25.something with 2 decimal places.

Checking the Options and Calculation

Let's examine the options provided and perform calculations to see which one, when squared, is equal to or very close to \(625.0025\). We will pay close attention to the calculation involving the option that is indicated as the correct answer.

The option suggested as correct is \(25.00005\). Let's square this number:

\((25.00005)^2 = 25.00005 \times 25.00005\)

We can calculate this directly or use the algebraic identity \((a+b)^2 = a^2 + 2ab + b^2\). Let \(a = 25\) and \(b = 0.00005\).

• \(a^2 = 25^2 = 625\)
• \(2ab = 2 \times 25 \times 0.00005 = 50 \times 0.00005\)

To calculate \(50 \times 0.00005\):

• \(b^2 = (0.00005)^2\)

To calculate \((0.00005)^2\):

\(0.00005\) has 5 decimal places. When squared, the result will have \(2 \times 5 = 10\) decimal places.

Now, combining the terms:

\((25.00005)^2 = a^2 + 2ab + b^2 = 625 + 0.0025 + 0.0000000025\)

\((25.00005)^2 = 625.0025000025\)

This result, \(625.0025000025\), is very close to the number \(625.0025\) given in the question. The difference is \(0.0000000025\).

Let's quickly check another option based on the decimal place rule, like \(25.05\). \((25.05)^2 = 25.05 \times 25.05\). We know \(2505^2 = 6275025\). Since \(25.05\) has 2 decimal places, \((25.05)^2\) will have \(2 \times 2 = 4\) decimal places. So, \((25.05)^2 = 627.5025\). This is not \(625.0025\).

Based on the calculation, squaring \(25.00005\) gives a value extremely close to \(625.0025\), suggesting that \(25.00005\) is the square root of a number very near to \(625.0025\).

Conclusion

By squaring the value \(25.00005\), we get \(625.0025000025\), which is a value very close to \(625.0025\).

Therefore, \(\sqrt{625.0025 }\) is approximately \(25.00005\).