Multiple choice

If $a\downarrow b = \sqrt [b]{a}$, then find the approximate value of $10\downarrow 3$.

  1. $1.12$
  2. $1.69$
  3. $2.15$
  4. $2.71$
  5. $3.33$
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C Correct answer
Explanation

The operation a downarrow b is defined as the b-th root of a. For 10 downarrow 3, we need the cube root of 10. Since 2^3 = 8 and 3^3 = 27, the value is slightly larger than 2, which is approximately 2.15.

AI explanation

The given operation a down arrow b defines the bth root of a, so 10 down arrow 3 means the cube root of 10. Use the linear approximation method with a nearby perfect cube, which is 8, having a cube root of 2. The derivative of the cube root function at 8 is 1 divided by 12, so the approximation is 2 plus (10 minus 8) divided by 12. This simplifies to 2 plus 1 divided by 6, yielding a result of approximately 2.15.

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