What is the approximate value of $\sqrt[5]{242.999}$?
- $\displaystyle\frac{1214999}{405000}$
- $\displaystyle\frac{1115}{405}$
- $\displaystyle\frac{121499}{40500}$
- $\displaystyle\frac{1214999}{4050}$
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AI explanation
Use the linear approximation formula for nth roots: the nth root of (A plus B) is approximately equal to the nth root of A plus B divided by (n times the nth root of A to the power of (n minus 1)). Let A be 243, B be negative 0.001, and n be 5, noting that the fifth root of 243 is 3. Substitute these values to get 3 plus negative 0.001 divided by the quantity 5 times 3 to the power of 4, which is 3 plus negative 0.001 divided by 405. Adjusting to a fraction with a common denominator gives 1215 divided by 405 minus 0.001 divided by 405, which equals 1214999 divided by 405000.