To rationalize the denominator of the expression \( \frac{\sqrt{72}}{\sqrt{7}} \), we can multiply both the numerator and the denominator by \( \sqrt{7} \) to eliminate the radical in the denominator.
Here are the steps:
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Multiply by \( \sqrt{7} \): \[ \frac{\sqrt{72}}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}} = \frac{\sqrt{72} \cdot \sqrt{7}}{7} \]
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Combine the radicals in the numerator: \[ = \frac{\sqrt{72 \cdot 7}}{7} \]
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Calculate \( 72 \cdot 7 \): \[ 72 \cdot 7 = 504 \] So, \[ = \frac{\sqrt{504}}{7} \]
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Simplify \( \sqrt{504} \): First, let's factor \( 504 \): \[ 504 = 72 \cdot 7 = 36 \cdot 14 = 36 \cdot (2 \cdot 7) = (6^2) \cdot (2 \cdot 7) = 36 \cdot 14 \]
However, we can take \( 504 \) apart a bit further: \[ 504 = 2^3 \cdot 3^2 \cdot 7 \] Taking the square root: \[ \sqrt{504} = \sqrt{(2^2 \cdot 3^2) \cdot (2 \cdot 7)} = \sqrt{36 \cdot 14} = 6\sqrt{14} \]
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Substituting back: \[ = \frac{6\sqrt{14}}{7} \]
Thus, the simplified form of the expression is: \[ \frac{6\sqrt{14}}{7} \]
Final answer: \[ \frac{6\sqrt{14}}{7} \]
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