Compute the square root of 37,221,835 to four decimals.

Compute the square root of 37,221,835 to four decimals.

• A. 60.9999
• B. 61.1097
• C. 61.1999
• D. 61.0097

Answer: (D)

Think of finding a square root as iteratively improving a guess. Start near a value whose square you know. Here, 6100 is a natural starting point because 6100^2 = 37,210,000, which is just below the target 37,221,835. Since the root should be a bit larger than 6100, you can use a Newton-Raphson update to refine it. Use the formula x1 = x0 + (N − x0^2) / (2x0). With x0 = 6100 and N = 37,221,835, you get: N − x0^2 = 11,835 and 2x0 = 12,200, so the correction is 11,835 / 12,200 ≈ 0.9700. Thus x1 ≈ 6100.9700. Squaring this value shows it’s already extremely close: (6100.9700)^2 ≈ 37,221,835.0 (to four decimals it matches the target). A tiny further refinement would place the value at about 6100.9700 to four decimals. So the square root to four decimals is 6100.9700. The options around 61 are off by roughly a factor of 100, suggesting a misprint in the choices.

Think of finding a square root as iteratively improving a guess. Start near a value whose square you know. Here, 6100 is a natural starting point because 6100^2 = 37,210,000, which is just below the target 37,221,835. Since the root should be a bit larger than 6100, you can use a Newton-Raphson update to refine it.

Use the formula x1 = x0 + (N − x0^2) / (2x0). With x0 = 6100 and N = 37,221,835, you get:

N − x0^2 = 11,835 and 2x0 = 12,200, so the correction is 11,835 / 12,200 ≈ 0.9700.

Thus x1 ≈ 6100.9700.

Squaring this value shows it’s already extremely close: (6100.9700)^2 ≈ 37,221,835.0 (to four decimals it matches the target). A tiny further refinement would place the value at about 6100.9700 to four decimals.

So the square root to four decimals is 6100.9700. The options around 61 are off by roughly a factor of 100, suggesting a misprint in the choices.