Estimate the square root to the nearest (a) integer and (b) tenth. \( \sqrt{46} \) a. nearest integer: \( \square \) b. nearest tenth: \( \square \)
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To estimate \( \sqrt{46} \), we start by identifying the perfect squares around 46. Since \( 6^2 = 36 \) and \( 7^2 = 49 \), we know that \( \sqrt{46} \) lies between 6 and 7. To narrow this down further, we can check the midpoint: \( 6.5^2 = 42.25 \) (too low), and \( 6.8^2 = 46.24 \) (too high). So, the nearest integer to \( \sqrt{46} \) is 7. To find the nearest tenth, let’s refine our estimate between 6.7 and 6.8. Since \( 6.7^2 = 44.89 \) and \( 6.8^2 = 46.24 \), you can tell that \( \sqrt{46} \) is closer to 6.8. Therefore, the nearest tenth is 6.8. a. nearest integer: 7 b. nearest tenth: 6.8