Find the 4th root of 82 by using differentials.

Question

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Calculator: 3.009217

 

Answer 1

y = f(x) = X^1/4.
dy/dx = 1 /(4 X^3/4)
dy = 1 / (4 X^3/4) dx --------------- (1)

dy is the change in y and dx is the change in x.

1⁴ = 1
2⁴ = 16
3⁴ =81

So 81= 3⁴ is close to 82.

Let x = 81

dx = 82-81 =1

Substitute 81 for x and 1 for dx in equuation (1).
dy = 1 / (4 X^3/4) dx

dy = 1 / (4 (81)^3/4) dx
     = 1 / (4 (3⁴)^3/4) (1)
     = 1 / (4 (3³))
     = 1 / ((4)(27))
     = 1 / 108
dy = 0.00926

Calculate the number
y + dy = (X^1/4) + (0.00926)

            =((81)^1/4) + (0.00926)

            =((3⁴)^1/4) + (0.00926)

            = 3 + 0.00926

            = 3.00926

Therefore, the approximate fourth root of 26 is 3.00926.