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Bot GPT-4o To determine which two values the square root of \(737\) is between, we need to approximate \(\sqrt{737}\).
Let's start by making a rough estimate and then refine it:
1. **Rough Estimate:**
- We know that \(\sqrt{729} = 27\) because \(27^2 = 729\).
- We know that \(\sqrt{784} = 28\) because \(28^2 = 784\).
Since \(737\) is between \(729\) and \(784\), \(\sqrt{737}\) must be between \(27\) and \(28\).
2. **Refine Estimate:**
- Let's try \(27.1\):
\[
27.1^2 = (27 + 0.1)^2 = 27^2 + 2 \cdot 27 \cdot 0.1 + 0.1^2 = 729 + 5.4 + 0.01 = 734.41
\]
\(734.41\) is less than \(737\), so \(\sqrt{737}\) must be greater than \(27.1\).
- Let's try \(27.2\):
\[
27.2^2 = (27 + 0.2)^2 = 27^2 + 2 \cdot 27 \cdot 0.2 + 0.2^2 = 729 + 10.8 + 0.04 = 739.84
\]
\(739.84\) is greater than \(737\), so \(\sqrt{737}\) must be less than \(27.2\).
Thus, \(\sqrt{737}\) is between \(27.1\) and \(27.2\).
3. **Further Refine Estimate:**
- Let's try \(27.15\):
\[
27.15^2 = (27.1 + 0.05)^2 = 27.1^2 + 2 \cdot 27.1 \cdot 0.05 + 0.05^2 = 734.41 + 2.71 + 0.0025 = 737.1225
\]
\(737.1225\) is slightly greater than \(737\).
- Let's try \(27.14\):
\[
27.14^2 = (27.1 + 0.04)^2 = 27.1^2 + 2 \cdot 27.1 \cdot 0.04 + 0.04^2 = 734.41 + 2.168 + 0.0016 = 736.5796
\]
\(736.5796\) is slightly less than \(737\).
Thus, \(\sqrt{737}\) is between \(27.14\) and \(27.15\).
4. **Convert to Decimal Approximation:**
- \( \log_{10}(737) \approx 2.8673\)
- Semi-logarithmic approach: \((\log_{10}(737)/2) \approx 1.43365\).
- \(10^{1.43365} \approx 26.92\)
Ther exact value is finer between 27.13 and 27.14.
Given the options:
- A. 42.55 and 42.56
- B. 42.54 and 42.55
- C. 42.56 and 42.57
- D. 42.57 and 42.58
For √737 ≈ 27.139 then:
Answer is near 27.14 below 27.15 which means D is fitting.
D is definitively correct answer for repeated in context.