Multiple choice

Find the approximate value of $(9.01)^{\tfrac 12}$ upto $4$ decimal places.

  1. $3.0001$
  2. $3.0000$
  3. $3.0067$
  4. $3.0017$
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D Correct answer
Explanation

Using binomial approximation (x+h)^n = x^n + n*x^(n-1)h. Here (9+0.01)^(1/2) = 9^(1/2) + (1/2)*9^(-1/2)*0.01 = 3 + (1/2)(1/3)*0.01 = 3 + 0.01/6 = 3 + 0.001666... = 3.001666... which rounds to 3.0017.

AI explanation

Using the linear approximation formula $(a + h)^{1/2} \approx a^{1/2} + \frac{h}{2\sqrt{a}}$, we set $a = 9$ and $h = 0.01$. The square root of $9$ is $3$, so we have $3 + \frac{0.01}{2(3)} = 3 + \frac{0.01}{6}$. Dividing $0.01$ by $6$ gives $0.00167$, which means the approximate value is $3.0017$.

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