To simplify the expression \( \sqrt{50} - 2\sqrt{8} \) and express it in the form \( 6\sqrt{x} \), we need to simplify each of the radicals.
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Simplifying \( \sqrt{50} \): \[ \sqrt{50} = \sqrt{25 \cdot 2} = \sqrt{25} \cdot \sqrt{2} = 5\sqrt{2} \]
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Simplifying \( \sqrt{8} \): \[ \sqrt{8} = \sqrt{4 \cdot 2} = \sqrt{4} \cdot \sqrt{2} = 2\sqrt{2} \]
Therefore: \[ 2\sqrt{8} = 2 \cdot 2\sqrt{2} = 4\sqrt{2} \]
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Combining the two simplified terms: Now we can substitute back into the original expression: \[ \sqrt{50} - 2\sqrt{8} = 5\sqrt{2} - 4\sqrt{2} = (5 - 4)\sqrt{2} = 1\sqrt{2} \]
Thus, the expression simplifies to: \[ \sqrt{2} \]
To express \( \sqrt{2} \) in the form \( 6\sqrt{x} \), we see that: \[ 1\sqrt{2} = 6\sqrt{\frac{2}{36}} = 6\sqrt{\frac{1}{18}} = 6\sqrt{2}\times \frac{1}{6} \] So, if we set \( 6\sqrt{x} = \sqrt{2} \), we must have \( x = \frac{1}{36} \).
Consequently, the original requested expression can be stated as: \[ \sqrt{50} - 2\sqrt{8} = 6\sqrt{\frac{1}{36}} \]
Thus, filling in the blanks: \[ \sqrt{50} - 2\sqrt{8} = 6\sqrt{2}, \text{, (so x = 2)} \] The final answer is:
- The value for \( x \) is \( 2 \).
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