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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 $19.6214$.
Answer for screen readers

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

Steps to Solve

  1. 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.

  2. 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: $$ \text{Midpoint} = \frac{19 + 20}{2} = 19.5 $$

  3. 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.

  4. 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.

  5. 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.
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