What is the square root of 85

General
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what is the square root of 85

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What is the square root of 85?

Answer: Finding the square root of 85 involves determining a number which, when multiplied by itself, gives the result of 85. The square root of 85 is not a perfect square, which means it cannot be expressed as a simple whole number. However, it can be approximated.

To find the square root of 85 more precisely, you can use a calculator or estimate it manually through various methods such as the long division method or using approximation techniques.

Using a Calculator

The most straightforward way is to use a calculator which will give:

\sqrt{85} \approx 9.219544

Manual Approximation Method

  1. Find nearby perfect squares:

    • The perfect square below 85 is 81 (since 9^2 = 81).
    • The perfect square above 85 is 100 (since 10^2 = 100).
    • Therefore, the square root of 85 lies between 9 and 10.

  2. Neighborhood examination:

    • Let’s refine our search. Since we know \sqrt{85} is a bit above 9 (closer to 81), we start testing values:

      For a finer approximation, let’s test 9.2 and 9.3:

      9.2^2 = 84.64
      9.3^2 = 86.49

      Since 84.64 is still closer to 85 but less when 84.64 then the root will be above 9.2 so we then next step further

  3. Further refinement:

    • Let’s test in smaller increments:
      • Check 9.22:
      9.22^2 = 85.0084
      Since 85.0084 is very close to 85, we see that \sqrt{85} \approx 9.22.

  4. Iterative approach:

    • You can continue this process, testing new values incrementally tighter until reaching the desired level of precision. Further refining, we find:
      \sqrt{85} \approx 9.219544

Conclusion

The square root of 85 is approximately 9.219544. In general, for most practical purposes, using a calculator provides a quick and accurate result.