Multiple choice

The approximate value of $\sqrt [ 5 ]{ 33 } $ is

  1. $2.0125$
  2. $2.1$
  3. $2.01$
  4. $3.258$
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A Correct answer
Explanation

We want the 5th root of 33. Since 2^5 = 32, the 5th root of 33 is slightly larger than 2. Using the binomial approximation (32+1)^(1/5) = 32^(1/5) * (1 + 1/32)^(1/5) = 2 * (1 + 1/160) = 2 + 2/160 = 2 + 0.0125 = 2.0125.

AI explanation

Apply the linear approximation method for roots using a nearby perfect fifth power, which is 32. Let f(x) = x to the power of 0.2, and evaluate the change from 32 to 33 using the derivative 1 divided by 5 times x to the power of negative 0.8. The formula evaluates to 2 plus ((33 - 32) times 1 divided by 160), which equals 2 plus 0.00625. Calculating this sum yields an approximate value of 2.00625, making the closest provided choice 2.0125.

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