To rationalize the denominator of the given ratio \( \frac{5}{\sqrt{34} + 2} \), we multiply both the numerator and the denominator by the conjugate of the denominator, which is \( \sqrt{34} - 2 \).
Here are the steps:
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Start with the expression: \[ \frac{5}{\sqrt{34} + 2} \]
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Multiply the numerator and denominator by \( \sqrt{34} - 2 \): \[ \frac{5(\sqrt{34} - 2)}{(\sqrt{34} + 2)(\sqrt{34} - 2)} \]
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Calculate the denominator using the difference of squares: \[ (\sqrt{34})^2 - (2)^2 = 34 - 4 = 30 \]
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Now, simplify the numerator: \[ 5(\sqrt{34} - 2) = 5\sqrt{34} - 10 \]
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Combine everything: \[ \frac{5\sqrt{34} - 10}{30} \]
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Simplify the fraction: \[ \frac{5\sqrt{34}}{30} - \frac{10}{30} = \frac{\sqrt{34}}{6} - \frac{1}{3} \]
Thus, the final expression for the ratio of the smaller tank to the larger one, with a rationalized denominator in simplified form, is: \[ \frac{5\sqrt{34} - 10}{30} \] Or in a compact form: \[ \frac{\sqrt{34}}{6} - \frac{1}{3} \] However, when presenting the complete rationalized form of the initial ratio \( \frac{5}{\sqrt{34} + 2} \), it's more straightforward to leave it in the form: \[ \frac{5(\sqrt{34} - 2)}{30} \] which can also be represented as: \[ \frac{\sqrt{34} - 2}{6} \]
Hence, the expression you would like is \( \frac{5\sqrt{34} - 10}{30} \) which maintains the ratio's original characteristics effectively while expressing it with rational numbers.
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