Question

Write this expression so that its denominator, if any, is a positive integer. \[\frac{2}{3 + \sqrt{7}}\].

Answer 1

Answer:

  3 -√7

Step-by-step explanation:

You want to rationalize the denominator of 2/(3 +√7).

Rationalize the denominator

The denominator of such an expression is turned to an integer by multiplying numerator and denominator by the conjugate of the denominator.

  $\dfrac{2}{3+\sqrt{7}}=\dfrac{2}{3+\sqrt{7}}\cdot\dfrac{3-\sqrt{7}}{3-\sqrt{7}}=\dfrac{2(3-\sqrt{7})}{3^2-7}=\boxed{3-\sqrt{7}}$

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Additional comment

This process takes advantage of the factorization of the difference of squares:

  a² -b² = (a +b)(a -b)

If 'b' is a value that can be made a rational real number by squaring it, then this multiplication by the conjugate can achieve that end. The conjugate of a sum is the corresponding difference.

This process is also used with complex numbers, since i² = -1, a real number.