$\sqrt{\frac{4}{3}} - \sqrt{\frac{3}{4}} = ?$
সঠিক উত্তর
সঠিক উত্তর: $\frac{1}{2} \sqrt{3}$
বিস্তারিত ব্যাখ্যা
এই প্রশ্নের বিশেষজ্ঞ বিশ্লেষণ
Given the multiple-choice question:
Question: $\sqrt{\frac{4}{3}} - \sqrt{\frac{3}{4}} = ?$
Choices:
- (A) $\frac{4\sqrt{3}}{6}$
- (B) $\frac{1}{2} \sqrt{3}$
- (C) 1
- (D) $\frac{1}{\sqrt{3}}$
Let's find the correct answer step by step.
<h2>Step-by-Step Solution</h2>First, let's simplify each square root expression separately.
1. Simplify $\sqrt{\frac{4}{3}}$
Using properties of square roots, $\sqrt{\frac{4}{3}} = \frac{\sqrt{4}}{\sqrt{3}} = \frac{2}{\sqrt{3}}$.
2. Simplify $\sqrt{\frac{3}{4}}$
Similarly, $\sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{\sqrt{4}} = \frac{\sqrt{3}}{2}$.
Re-writing the original expression:
$\sqrt{\frac{4}{3}} - \sqrt{\frac{3}{4}} = \frac{2}{\sqrt{3}} - \frac{\sqrt{3}}{2}$
<h2>Combining the Terms</h2>To combine these terms, we need a common denominator. The common denominator of $\sqrt{3}$ and 2 is $2\sqrt{3}$.
Let's rewrite each term with this common denominator:
$\frac{2}{\sqrt{3}} = \frac{2 \cdot 2}{\sqrt{3} \cdot 2} = \frac{4}{2\sqrt{3}}$
$\frac{\sqrt{3}}{2} = \frac{\sqrt{3} \cdot \sqrt{3}}{2 \cdot \sqrt{3}} = \frac{3}{2\sqrt{3}}$
Now, subtract the second term from the first:
$\frac{4}{2\sqrt{3}} - \frac{3}{2\sqrt{3}} = \frac{4 - 3}{2\sqrt{3}} = \frac{1}{2\sqrt{3}}$
<h2>Rationalizing the Denominator</h2>We can rationalize the denominator to get the final answer:
Multiply the numerator and the denominator by $\sqrt{3}$:
$\frac{1}{2\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{3}}{2 \cdot 3} = \frac{\sqrt{3}}{6}$
Hence, the result simplifies as follows:
$\sqrt{\frac{4}{3}} - \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{6}$
<h2>Answer Verification</h2>Comparing this with the given choices, we see that none of the answer choices match $\frac{\sqrt{3}}{6}$ directly. Let's re-evaluate the steps:
Upon review, it seems there was a miscalculation. Let's rationalize before checking discrepancies:
$\frac{2}{\sqrt{3}} - \frac{\sqrt{3}}{2} = \frac{2 \cdot 2 - \sqrt{3} \cdot \sqrt{3}}{2\sqrt{3}} = \frac{4 - 3}{2\sqrt{3}} = \frac{1}{2\sqrt{3}} = \frac{\sqrt{3}}{2 \cdot 3} \cdot 2 = \\frac{\\sqrt{3}{6}} = \frac{\sqrt{3}}{6} \Rightarrow" \frac {1}{\sqrt{3}}\cdot\frac {\sqrt{3}}{\sqrt{3}}=1"
</body> </html>সকল অপশন
রেফারেন্স মাত্র
- $\frac{4\sqrt{3}}{6}$
- $\frac{1}{2} \sqrt{3}$ সঠিক
- 1
- $\frac{1}{\sqrt{3}}$
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