Questions: Rationalize the denominator of the fraction below. What is the new denominator? 4/(3+√7) A. 16 B. -40 C. -4 D. 2

Solution Steps

To rationalize the denominator of the fraction \(\frac{4}{3+\sqrt{7}}\), we multiply both the numerator and the denominator by the conjugate of the denominator, which is \(3-\sqrt{7}\). This will eliminate the square root in the denominator.

To rationalize the denominator of the fraction \(\frac{4}{3+\sqrt{7}}\), we first identify the conjugate of the denominator. The conjugate of \(3+\sqrt{7}\) is \(3-\sqrt{7}\).

We multiply both the numerator and the denominator of the fraction by the conjugate \(3-\sqrt{7}\):

\[ \frac{4}{3+\sqrt{7}} \times \frac{3-\sqrt{7}}{3-\sqrt{7}} = \frac{4(3-\sqrt{7})}{(3+\sqrt{7})(3-\sqrt{7})} \]

The new numerator becomes:

\[ 4(3-\sqrt{7}) = 12 - 4\sqrt{7} \]

The new denominator is a difference of squares:

\[ (3+\sqrt{7})(3-\sqrt{7}) = 3^2 - (\sqrt{7})^2 = 9 - 7 = 2 \]

Final Answer