Find the square root of 95 by estimation method.

Find the square root of 95 by estimation method. Give a full explanation…

The Correct Answer and Explanation is:

To find the square root of 95 by estimation, we use the estimation method which involves identifying two perfect squares between which 95 falls and then refining our guess.

Step-by-step estimation:

1. Identify nearby perfect squares:
   • 81=9\sqrt{81} = 981​=9
   • 100=10\sqrt{100} = 10100​=10
   Since 81<95<10081 < 95 < 10081<95<100, it means: 95 is between 9 and 10\sqrt{95} \text{ is between } 9 \text{ and } 1095​ is between 9 and 10
2. Try the midpoint between 9 and 10: Try 9.59.59.5 9.5×9.5=90.25(Too low)9.5 \times 9.5 = 90.25 \quad \text{(Too low)}9.5×9.5=90.25(Too low)
3. Try a higher value, like 9.7: 9.7×9.7=94.09(Still slightly low)9.7 \times 9.7 = 94.09 \quad \text{(Still slightly low)}9.7×9.7=94.09(Still slightly low)
4. Try 9.8: 9.8×9.8=96.04(Slightly high)9.8 \times 9.8 = 96.04 \quad \text{(Slightly high)}9.8×9.8=96.04(Slightly high)
5. Try 9.75: 9.75×9.75=95.06(Very close)9.75 \times 9.75 = 95.06 \quad \text{(Very close)}9.75×9.75=95.06(Very close) So, 95≈9.75\sqrt{95} \approx 9.7595​≈9.75

Answer:

95≈9.75\boxed{\sqrt{95} \approx 9.75}95​≈9.75​

Explanation

The square root of a number is a value that, when multiplied by itself, gives the original number. Since 95 is not a perfect square, we can estimate its square root using numbers whose squares we already know.

We start by identifying the two perfect squares that are closest to 95. We find that 81 and 100 are perfect squares and that 95 falls between them. The square root of 81 is 9, and the square root of 100 is 10. This means the square root of 95 lies somewhere between 9 and 10.

To refine the estimate, we test values between 9 and 10 by squaring them. Trying 9.5 gives us 90.25, which is too low. Trying 9.7 gives us 94.09, which is still below 95 but closer. Then, trying 9.8 gives us 96.04, which is slightly above 95. That tells us the square root of 95 is between 9.7 and 9.8.

To get even closer, we try 9.75. Squaring 9.75 gives us 95.06, which is very close to 95. This indicates that 9.75 is a strong estimate for the square root of 95.

This method works well when a calculator is not allowed or when you need to estimate quickly using mental math or simple arithmetic. The more decimal places you test, the closer you get to the actual square root.