square root of 385

Understand the Problem

The question is asking for the calculation of the square root of 385, which is a mathematical operation. To find the square root, we look for a number that, when multiplied by itself, equals 385.

Answer

The square root of 385 is approximately $19.6214$.

Steps to Solve

Estimate the square root To find the square root of 385, we can start by determining two numbers whose squares are close to 385. We know that $19^2 = 361$ and $20^2 = 400$. Thus, the square root of 385 is between 19 and 20.
Refine the estimate using averaging To get a more accurate value, we can average our approximate square root. Take the midpoint of 19 and 20: $$ \text{Midpoint} = \frac{19 + 20}{2} = 19.5 $$
Check the square of the average Next, we calculate $19.5^2$ to see how it compares to 385: $$ 19.5^2 = 380.25 $$ Since $380.25 < 385$, we know the square root is greater than 19.5.
Further refining with another average Now let's try a higher estimate, say 19.6: $$ 19.6^2 = 384.16 $$ This is still less than 385, so we try an even higher number, like 19.7: $$ 19.7^2 = 388.09 $$ Now it's greater than 385, suggesting that the true value lies between 19.6 and 19.7.
Final approximation using a calculator Using a calculator to find $\sqrt{385}$ gives approximately $19.6214$. This value also fits within our established bounds.

The square root of 385 is approximately $\sqrt{385} \approx 19.6214$.

More Information

The square root of a number represents a value that, when multiplied by itself, gives the original number. This can be useful in various applications like geometry, physics, and engineering. The square root of 385 is not a whole number, indicating it lies between two integers.

Tips

Not checking the bounds correctly: Always ensure that your approximations stay within the established limits.
Forgetting to square the estimated value: It’s important to verify by squaring your current estimate to confirm how close it is to the original number.

Thank you for voting!