Approximate value of $\sqrt[4]{81.04}$
- $3.0003$
- $3.004$
- $3.0046$
- $3.0012$
Reveal answer
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A
Correct answer
Explanation
Using binomial approximation (x+dx)^(1/n) approx x^(1/n) + (1/n)x^((1/n)-1)*dx. Here x=81, dx=0.04, n=4. 81^(1/4) = 3. Approximation = 3 + (1/4)*81^(-3/4)*0.04 = 3 + (1/4)(1/27)*0.04 = 3 + 0.01/27 = 3 + 0.00037. The closest option is 3.0003.
AI explanation
Using linear approximation for roots, the fourth root of 81.04 can be estimated by finding the fourth root of 81 and adjusting for the small difference of 0.04. Since the fourth root of 81 is exactly 3, the formula f(x) plus f prime of x multiplied by delta x is used, where f prime of x is one divided by four times x cubed. This gives an adjustment of 0.04 divided by 108, which equals 0.00037. Adding this to 3 gives the approximate result of 3.0003.