Verified Answer :

  • Identify the conjugate of the denominator, \(7 + \sqrt{3}\).
  • Multiply both numerator and denominator by the conjugate to rationalize the expression.
  • Apply the difference of squares identity, \((a - b)(a + b) = a^2 - b^2\), to simplify the denominator to \(46\).
  • Simplify the final fraction \(\frac{14 + 2\sqrt{3}}{46}\) by dividing by 2, resulting in \(\boxed{\frac{7 + \sqrt{3}}{23}}\).

Explanation

  1. Problem Analysis and Objective

We are given the expression \(\frac{2}{7 - \sqrt{3}}\) and we need to rationalize its denominator. Rationalizing the denominator means rewriting the fraction so that the denominator is a rational number. To do this, we use the algebraic property of the difference of two squares: \((a - b)(a + b) = a^2 - b^2\). This identity allows us to eliminate a square root by multiplying a binomial of the form \(a - \sqrt{b}\) by its conjugate \(a + \sqrt{b}\).

  1. Multiplying by the Conjugate

In the expression \(\frac{2}{7 - \sqrt{3}}\), the denominator is \(7 - \sqrt{3}\). The conjugate of this denominator is \(7 + \sqrt{3}\). By multiplying both the numerator and the denominator of the original fraction by this conjugate, we can rationalize the denominator without changing the value of the expression. This gives us:
\[\frac{2}{7 - \sqrt{3}} \times \frac{7 + \sqrt{3}}{7 + \sqrt{3}} = \frac{2(7 + \sqrt{3})}{(7 - \sqrt{3})(7 + \sqrt{3})}\]

  1. Simplifying the Denominator and Numerator

Now we simplify the numerator and the denominator separately. The numerator is \(2(7 + \sqrt{3}) = 14 + 2\sqrt{3}\). For the denominator, we use the identity \((a - b)(a + b) = a^2 - b^2\):
\[(7 - \sqrt{3})(7 + \sqrt{3}) = 7^2 - (\sqrt{3})^2 = 49 - 3 = 46.\]
Substituting these results back into our expression, we get:
\[\frac{14 + 2\sqrt{3}}{46}\]

  1. Final Simplification

To reach the simplest form, we divide both the numerator and the denominator by their greatest common divisor, which is 2:
\[\frac{14 + 2\sqrt{3}}{46} = \frac{2(7 + \sqrt{3})}{46} = \frac{7 + \sqrt{3}}{23}.\]
Thus, the rationalized form of the expression is \(\frac{7 + \sqrt{3}}{23}\).

Examples

Rationalizing denominators is a fundamental technique in algebra that simplifies expressions and makes them easier to work with, especially when performing further calculations or comparing values. For instance, in engineering or physics, when dealing with complex impedances or electrical circuit formulas containing square roots, converting to a rational denominator helps in standardizing the form, leading to cleaner results for subsequent steps in problem-solving. This ensures consistency and precision across mathematical and physical models.

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