Multiple choice

Using the differentials, the approximate value of $(627)^{1/4}$ is

  1. 5.002

  2. 5.003

  3. 5.005

  4. 5.004
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D Correct answer
Explanation

Using differentials, f(x+dx) is approximately f(x) + f'(x)dx. For (627)^(1/4), let x = 625 and dx = 2. Then 625^(1/4) = 5, and f'(x) = (1/4)x^(-3/4). Calculating 5 + (1/4)(625^(-3/4))(2) gives 5 + 0.5/125 = 5 + 0.004 = 5.004.

AI explanation

Using the method of differentials, we approximate the fourth root of 625 plus 2 by using the function x to the power of 1/4. The derivative is 1 divided by 4 times x to the power of 3/4, and evaluating this at 625 gives 1 divided by 4 times 125, which is 1/500. We multiply this derivative by 2 to get a change of 0.004, and adding this to the fourth root of 625, which is 5, yields 5.004.

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