To estimate the sum \( \sqrt{5} + \sqrt{7} \) with increasingly small rational approximations, let's first approximate each square root.
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Estimate \( \sqrt{5} \) and \( \sqrt{7} \):
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\( \sqrt{5} \) is between \( 2.2 \) and \( 2.3 \). Calculating:
- \( 2.2^2 = 4.84 \)
- \( 2.3^2 = 5.29 \)
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Thus, \( \sqrt{5} \approx 2.236 \).
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\( \sqrt{7} \) is between \( 2.6 \) and \( 2.7 \):
- \( 2.6^2 = 6.76 \)
- \( 2.7^2 = 7.29 \)
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Thus, \( \sqrt{7} \approx 2.645 \).
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Adding the approximations: \[ \sqrt{5} + \sqrt{7} \approx 2.236 + 2.645 = 4.881. \]
This initial estimate gives us \( 4.881 \).
To improve the estimate, we can use more decimal places for the square roots:
- Using a calculator, we find:
- \( \sqrt{5} \approx 2.236067977 \)
- \( \sqrt{7} \approx 2.645751311 \)
Adding these more accurate values: \[ \sqrt{5} + \sqrt{7} \approx 2.236067977 + 2.645751311 \approx 4.881819288. \]
Rounding this to four decimal places gives us \( 4.8818 \).
Thus, the answer is: \[ \boxed{4.8818} \]