Question

$\sqrt{6777}$

Answer 1

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The square root of 6777 is 82.322536404073. Or,  

√6777 = 82.322536404073

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Step 1:  

Divide the number (6777) by 2 to get the first guess for the square root .

First guess = 6777/2 = 3388.5.

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Step 2:

Divide 6777 by the previous result. d = 6777/3388.5 = 2.

Average this value (d) with that of step 1: (2 + 3388.5)/2 = 1695.25 (new guess).

Error = new guess - previous value = 3388.5 - 1695.25 = 1693.25.

1693.25 > 0.001. As error > accuracy, we repeat this step again.

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Step 3:

Divide 6777 by the previous result. d = 6777/1695.25 = 3.997640466.

Average this value (d) with that of step 2: (3.997640466 + 1695.25)/2 = 849.623820233 (new guess).

Error = new guess - previous value = 1695.25 - 849.623820233 = 845.626179767.

845.626179767 > 0.001. As error > accuracy, we repeat this step again.

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Step 4:

Divide 6777 by the previous result. d = 6777/849.623820233 = 7.9764712789.

Average this value (d) with that of step 3: (7.9764712789 + 849.623820233)/2 = 428.800145756 (new guess).

Error = new guess - previous value = 849.623820233 - 428.800145756 = 420.823674477.

420.823674477 > 0.001. As error > accuracy, we repeat this step again.

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Step 5:

Divide 6777 by the previous result. d = 6777/428.800145756 = 15.8045655233.

Average this value (d) with that of step 4: (15.8045655233 + 428.800145756)/2 = 222.3023556397 (new guess).

Error = new guess - previous value = 428.800145756 - 222.3023556397 = 206.4977901163.

206.4977901163 > 0.001. As error > accuracy, we repeat this step again.

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Step 6:

Divide 6777 by the previous result. d = 6777/222.3023556397 = 30.4855069147.

Average this value (d) with that of step 5: (30.4855069147 + 222.3023556397)/2 = 126.3939312772 (new guess).

Error = new guess - previous value = 222.3023556397 - 126.3939312772 = 95.9084243625.

95.9084243625 > 0.001. As error > accuracy, we repeat this step again.

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Step 7:

Divide 6777 by the previous result. d = 6777/126.3939312772 = 53.6180806429.

Average this value (d) with that of step 6: (53.6180806429 + 126.3939312772)/2 = 90.0060059601 (new guess).

Error = new guess - previous value = 126.3939312772 - 90.0060059601 = 36.3879253171.

36.3879253171 > 0.001. As error > accuracy, we repeat this step again.

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Step 8:

Divide 6777 by the previous result. d = 6777/90.0060059601 = 75.2949753487.

Average this value (d) with that of step 7: (75.2949753487 + 90.0060059601)/2 = 82.6504906544 (new guess).

Error = new guess - previous value = 90.0060059601 - 82.6504906544 = 7.3555153057.

7.3555153057 > 0.001. As error > accuracy, we repeat this step again.

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Step 9:

Divide 6777 by the previous result. d = 6777/82.6504906544 = 81.9958834647.

Average this value (d) with that of step 8: (81.9958834647 + 82.6504906544)/2 = 82.3231870596 (new guess).

Error = new guess - previous value = 82.6504906544 - 82.3231870596 = 0.3273035948.

0.3273035948 > 0.001. As error > accuracy, we repeat this step again.

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Step 10:

Divide 6777 by the previous result. d = 6777/82.3231870596 = 82.3218857537.

Average this value (d) with that of step 9: (82.3218857537 + 82.3231870596)/2 = 82.3225364067 (new guess).

Error = new guess - previous value = 82.3231870596 - 82.3225364067 = 0.0006506529.

0.0006506529 <= 0.001. As error <= accuracy, we stop the iterations and use 82.3225364067 as the square root.

So, we can say that the square root of 6777 is 82.322 with an error smaller than 0.001 (in fact the error is 0.0006506529). this means that the first 3 decimal places are correct. Just to compare, the returned value by using the javascript function 'Math.sqrt(6777)' is 82.32253640407346.

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Hope it helps you dear.

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