To estimate the sum of \(\sqrt{5} + \sqrt{7}\), we can find the square roots of 5 and 7 using increasingly smaller rational approximations.
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Estimate \(\sqrt{5}\):
- We know that \(2^2 = 4\) and \(3^2 = 9\), so \(\sqrt{5}\) is between 2 and 3.
- To get more precise, we can try numbers between 2.2 and 2.3:
- \(2.2^2 = 4.84\)
- \(2.3^2 = 5.29\)
Thus, \(\sqrt{5}\) is between 2.2 and 2.3.
Next, try \(2.24\):
- \(2.24^2 = 5.0176\) (too high)
- Next, try \(2.23\):
- \(2.23^2 = 4.9729\) (too low)
So \(\sqrt{5}\) is between \(2.23\) and \(2.24\). We can take the average: \[ \sqrt{5} \approx 2.236 \]
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Estimate \(\sqrt{7}\):
- We know that \(2.6^2 = 6.76\) and \(2.7^2 = 7.29\), so \(\sqrt{7}\) is between 2.6 and 2.7.
- Next, try \(2.65\):
- \(2.65^2 = 7.0225\) (too high)
- Next, try \(2.64\):
- \(2.64^2 = 6.9696\) (too low)
Thus, we find \(\sqrt{7}\) is between \(2.64\) and \(2.65\). Using the average: \[ \sqrt{7} \approx 2.645 \]
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Sum the Approximations: \[ \sqrt{5} + \sqrt{7} \approx 2.236 + 2.645 = 4.881 \]
Using a calculator or more precise approximations, we find that:
- \(\sqrt{5} \approx 2.2361\)
- \(\sqrt{7} \approx 2.6458\)
Therefore: \[ \sqrt{5} + \sqrt{7} \approx 4.8819 \]
Based on the options given, the best approximation to four decimal places is: 4.8819