- Identify the denominator \(\sqrt{14}-\sqrt{3}\) and its conjugate \(\sqrt{14}+\sqrt{3}\).
- Multiply both numerator and denominator by the conjugate: \(\frac{1(\sqrt{14}+\sqrt{3})}{(\sqrt{14}-\sqrt{3})(\sqrt{14}+\sqrt{3})}\).
- Simplify the denominator using the difference of squares: \(14 - 3 = 11\).
- Match the resulting expression \(\frac{\sqrt{14}+\sqrt{3}}{11}\) to the target to find \([?] = 14\).
- The final result is \(\boxed{14}\).
Explanation
- Analysis of the Radical Expression
To simplify the radical expression \(\frac{1}{\sqrt{14}-\sqrt{3}}\), we need to rationalize the denominator. The denominator is in the form \(\sqrt{a} - \sqrt{b}\), where \(a = 14\) and \(b = 3\). To eliminate the square roots in the denominator, we multiply the numerator and the denominator by the conjugate of the denominator, which is \(\sqrt{14} + \sqrt{3}\). This approach relies on the difference of squares identity, \((x-y)(x+y) = x^2 - y^2\), which will remove the square roots once the terms are squared.
- Simplifying the Expression
We multiply the fraction \(\frac{1}{\sqrt{14}-\sqrt{3}}\) by \(\frac{\sqrt{14}+\sqrt{3}}{\sqrt{14}+\sqrt{3}}\):
\[ \frac{1}{\sqrt{14}-\sqrt{3}} \cdot \frac{\sqrt{14}+\sqrt{3}}{\sqrt{14}+\sqrt{3}} = \frac{\sqrt{14}+\sqrt{3}}{(\sqrt{14}-\sqrt{3})(\sqrt{14}+\sqrt{3})} \]
The denominator simplifies using the identity \((a-b)(a+b) = a^2 - b^2\):
\[ (\sqrt{14})^2 - (\sqrt{3})^2 = 14 - 3 = 11 \]
Thus, the expression becomes:
\[ \frac{\sqrt{14}+\sqrt{3}}{11} \]
Comparing this to the original target expression \(\frac{\sqrt{[?]}+\sqrt{3}}{11}\), we can see that \([?] = 14\).
- Conclusion
By rationalizing the denominator, we found that \(\frac{1}{\sqrt{14}-\sqrt{3}} = \frac{\sqrt{14}+\sqrt{3}}{11}\). Therefore, the value in the bracket is \(\boxed{14}\).
Examples
Rationalizing denominators is a fundamental technique in algebra and calculus, often used to simplify limits, solve equations, and perform complex calculations. In real life, this is relevant in fields like engineering and physics, where engineers may need to manipulate formulas involving square roots to calculate stress, strain, or energy levels efficiently. By converting a fraction with a radical denominator into a simpler form, calculations become much more straightforward and less prone to error.