Question

find the value of √7.9
with step by step explanation​

Answer 1

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$\color{red}\huge{\underline{\underline{\mathfrak{Question:-}}}}$

Find the value of $\sqrt{7.9}$

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$\color{blue}\huge{\underline{\underline{\mathfrak{Answer}}}}$

${\blue{\bold{2.8106938645}}}$

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$\color{orange}\huge{\underline{\underline{\mathfrak{Solution:-}}}}$

${\bold{\pink{\text{We will solve it by Hero's Method}}}}$

${\bold{\green{\text{Step 1 :-}}}}$

 Divide the number (7.9) by 2 to get the first guess for the square root .

 First guess = 7.9/2 = 3.95.

${\bold{\green{\text{Step 2 :-}}}}$

 Divide 7.9 by the previous result. d = 7.9/3.95 = 2.

 Average this value (d) with that of step 1: (2 + 3.95)/2 = 2.975 (new guess).

 Error = new guess - previous value = 3.95 - 2.975 = 0.975.

 0.975 > 0.001. As error > accuracy, we repeat this step again.

${\bold{\green{\text{Step 3 :-}}}}$

 Divide 7.9 by the previous result. d = 7.9/2.975 = 2.6554621849.

 Average this value (d) with that of step 2: (2.6554621849 + 2.975)/2 = 2.8152310925 (new guess).

 Error = new guess - previous value = 2.975 - 2.8152310925 = 0.1597689075.

 0.1597689075 > 0.001. As error > accuracy, we repeat this step again.

${\bold{\green{\text{Step 4 :-}}}}$

 Divide 7.9 by the previous result. d = 7.9/2.8152310925 = 2.806163949.

 Average this value (d) with that of step 3: (2.806163949 + 2.8152310925)/2 = 2.8106975207 (new guess).

 Error = new guess - previous value = 2.8152310925 - 2.8106975207 = 0.0045335718.

 0.0045335718 > 0.001. As error > accuracy, we repeat this step again.

${\bold{\green{\text{Step 5 :-}}}}$

 Divide 7.9 by the previous result. d = 7.9/2.8106975207 = 2.8106902083.

 Average this value (d) with that of step 4: (2.8106902083 + 2.8106975207)/2 = 2.8106938645 (new guess).

 Error = new guess - previous value = 2.8106975207 - 2.8106938645 = 0.0000036562.

 0.0000036562 <= 0.001. As error <= accuracy, we stop the iterations and use 2.8106938645 as the square root.

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${\bold{\red{\text{What is Hero's Method? }}}}$

$\implies$ An iterative method of approximating the square root of a number. If √k is required, and x 0 is an initial approximation, then n = 0, 1, 2, … will converge to the square root of k. For example, to calculate the square root of 5, using a first approximation of 2, will give x 2=2.236  111  11…, x 3=2.236  067  978…, x 4=2.236  067  978… and √5 = 2.236067978…. So this method has found the square root to considerable accuracy after only three iterations.

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Hope it helps