To determine which of the provided options is an irrational number between 4 and 8, let's analyze each choice:
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\(5\sqrt{}\): This seems to imply \( \sqrt{5} \). We will check its value:
- \( \sqrt{5} \approx 2.236\) (this is less than 4, so it does not meet the condition).
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5: This is a rational number, as it can be expressed as \( \frac{5}{1} \).
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Square root of 5 (\( \sqrt{5} \)): As calculated before, \( \sqrt{5} \approx 2.236\) (again this is less than 4, so it does not meet the condition).
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\(8\sqrt{}\): This seems to imply \( \sqrt{8} \). Let's find its value:
- \( \sqrt{8} = 2\sqrt{2} \approx 2 \times 1.414 = 2.828 \) (still less than 4).
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8: This is also a rational number, as it can be expressed as \( \frac{8}{1} \).
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Square root of 8 (\( \sqrt{8} \)): As calculated before, this is \( 2.828\), which is less than 4.
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\(12\sqrt{}\): This seems to imply \( \sqrt{12} \). Let's find its value:
- \( \sqrt{12} = 2\sqrt{3} \) and \( \sqrt{3} \approx 1.732\) so \( 2\sqrt{3} \approx 2 \times 1.732 = 3.464 \) (which is still less than 4).
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12: This is a rational number, as it can also be expressed as \( \frac{12}{1} \).
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\(17\sqrt{}\): This seems to imply \( \sqrt{17} \). Let's find its value:
- \( \sqrt{17} \approx 4.123 \) (which is between 4 and 8 and is an irrational number).
So the only irrational number between 4 and 8 is \( \sqrt{17} \).