To determine whether the expression \(\sqrt{\frac{47}{3} + 121}\) produces a rational or irrational number, we will first simplify the expression inside the square root.
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First, convert \(121\) into a fraction with a denominator of \(3\): \[ 121 = \frac{121 \times 3}{3} = \frac{363}{3} \]
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Next, add \(\frac{47}{3}\) and \(\frac{363}{3}\): \[ \frac{47}{3} + \frac{363}{3} = \frac{47 + 363}{3} = \frac{410}{3} \]
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Now, we can write the expression under the square root: \[ \sqrt{\frac{410}{3}} \]
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We can further simplify this: \[ \sqrt{\frac{410}{3}} = \frac{\sqrt{410}}{\sqrt{3}} \]
To determine if \(\sqrt{\frac{410}{3}}\) is rational or irrational, we need to investigate \(\sqrt{410}\) and \(\sqrt{3}\):
- \(\sqrt{3}\) is known to be an irrational number.
- For \(\sqrt{410}\), we need to check if \(410\) is a perfect square. The prime factorization of \(410\) is: \[ 410 = 2 \times 5 \times 41 \] Since \(410\) does not have any factor raised to an even power, \(\sqrt{410}\) is also an irrational number.
Since the product of two irrational numbers (as long as they are not precisely structured to yield a rational product) will typically result in an irrational number, and given that both \(\sqrt{410}\) and \(\sqrt{3}\) are irrational, we conclude:
\[ \sqrt{\frac{410}{3}} = \frac{\sqrt{410}}{\sqrt{3}} \text{ is also irrational.} \]
Thus, the expression \(\sqrt{\frac{47}{3} + 121}\) produces an irrational number.