The problem asks to express the fraction $\frac{3\sqrt{2} - \sqrt{3}}{2\sqrt{3} - \sqrt{2}}$ in the form $\frac{\sqrt{m}}{\sqrt{n}}$, where $m$ and $n$ are whole numbers.

AlgebraSimplificationRadicalsRationalizationFractionsExponents
2025/3/19

1. Problem Description

The problem asks to express the fraction 323232\frac{3\sqrt{2} - \sqrt{3}}{2\sqrt{3} - \sqrt{2}} in the form mn\frac{\sqrt{m}}{\sqrt{n}}, where mm and nn are whole numbers.

2. Solution Steps

First, we multiply the numerator and denominator by the conjugate of the denominator, which is 23+22\sqrt{3} + \sqrt{2}:
32323223+223+2=(323)(23+2)(232)(23+2) \frac{3\sqrt{2} - \sqrt{3}}{2\sqrt{3} - \sqrt{2}} \cdot \frac{2\sqrt{3} + \sqrt{2}}{2\sqrt{3} + \sqrt{2}} = \frac{(3\sqrt{2} - \sqrt{3})(2\sqrt{3} + \sqrt{2})}{(2\sqrt{3} - \sqrt{2})(2\sqrt{3} + \sqrt{2})}
Now, we expand the numerator and denominator:
Numerator:
(323)(23+2)=32(23)+32(2)3(23)3(2)=66+3(2)2(3)6=66+666=56(3\sqrt{2} - \sqrt{3})(2\sqrt{3} + \sqrt{2}) = 3\sqrt{2}(2\sqrt{3}) + 3\sqrt{2}(\sqrt{2}) - \sqrt{3}(2\sqrt{3}) - \sqrt{3}(\sqrt{2}) = 6\sqrt{6} + 3(2) - 2(3) - \sqrt{6} = 6\sqrt{6} + 6 - 6 - \sqrt{6} = 5\sqrt{6}
Denominator:
(232)(23+2)=(23)2(2)2=4(3)2=122=10(2\sqrt{3} - \sqrt{2})(2\sqrt{3} + \sqrt{2}) = (2\sqrt{3})^2 - (\sqrt{2})^2 = 4(3) - 2 = 12 - 2 = 10
So we have:
5610=62 \frac{5\sqrt{6}}{10} = \frac{\sqrt{6}}{2}
Now, we want to express 62\frac{\sqrt{6}}{2} in the form mn\frac{\sqrt{m}}{\sqrt{n}}.
Since 2=42 = \sqrt{4}, we can write:
62=64=64=32 \frac{\sqrt{6}}{2} = \frac{\sqrt{6}}{\sqrt{4}} = \sqrt{\frac{6}{4}} = \sqrt{\frac{3}{2}}
To avoid a fraction inside the square root, we can manipulate the original expression further:
62=6444=644=244=2416 \frac{\sqrt{6}}{2} = \frac{\sqrt{6}}{\sqrt{4}} \cdot \frac{\sqrt{4}}{\sqrt{4}} = \frac{\sqrt{6\cdot4}}{4} = \frac{\sqrt{24}}{4} = \frac{\sqrt{24}}{\sqrt{16}}
This form satisfies the requirement of having whole numbers under the square root. However, looking at the given options, we see that we can directly write
5610=256100=150100 \frac{5\sqrt{6}}{10} = \frac{\sqrt{25 \cdot 6}}{\sqrt{100}} = \frac{\sqrt{150}}{\sqrt{100}}
So, m=150m = 150 and n=100n = 100.

3. Final Answer

The final answer is (b) 150100\frac{\sqrt{150}}{\sqrt{100}}.

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