To determine which of the given square roots is an irrational number between 8 and 10, we’ll calculate the approximate values of \( \sqrt{50} \), \( \sqrt{84} \), \( \sqrt{62} \), and \( \sqrt{120} \).
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Calculate \( \sqrt{50} \): \[ \sqrt{50} \approx 7.07 \quad (\text{since } 7^2 = 49 \text{ and } 8^2 = 64) \] Since \( 7.07 \) is less than \( 8 \), \( \sqrt{50} \) is not between \( 8 \) and \( 10 \).
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Calculate \( \sqrt{84} \): \[ \sqrt{84} \approx 9.17 \quad (\text{since } 9^2 = 81 \text{ and } 10^2 = 100) \] \( 9.17 \) is between \( 8 \) and \( 10 \).
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Calculate \( \sqrt{62} \): \[ \sqrt{62} \approx 7.87 \quad (\text{since } 7^2 = 49 \text{ and } 8^2 = 64) \] Since \( 7.87 \) is less than \( 8 \), \( \sqrt{62} \) is not between \( 8 \) and \( 10 \).
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Calculate \( \sqrt{120} \): \[ \sqrt{120} \approx 10.95 \quad (\text{since } 10^2 = 100 \text{ and } 11^2 = 121) \] Since \( 10.95 \) is greater than \( 10 \), \( \sqrt{120} \) is not between \( 8 \) and \( 10 \).
Conclusion:
Out of the evaluated options, \( \sqrt{84} \) is an irrational number between \( 8 \) and \( 10 \).