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The N-R method is an iterative method whereby:
x{n+!} = x{n} - f(x{n})/f'(x{n})
and repeat until |x{n+1} - x{n}| < ε for some small number ε which gives the level of accuracy required.
In this case: f(x) = x³ - 12
→ f'(x) = 3x²
→ x{n+1} = x{n} - (x{n}³-12)/(3x{n}²)
The cube root of 12 lies between 2 (2³ = 8) and 3 (3³ = 27), so pick a value close to 2, say 2.1 as an initial guess then:
x{0} = 2.1
→ x{1} = 2.1 - (2.1³ - 12)/(3×2.1²) ≈ 2.3070
→ x{2} = 2.3070 - (2.3070³ - 12)/(3×2.3070²) ≈ 2.2896
and so on.
with this |x{7} - x{6}| < 1×10^-10 → cube root 12 to 9 dp is 2.289428485.
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