Problem 49 - College Algebra

Question

Rationalize the denominator. $$\frac{13}{3+\sqrt{11}}$$

Step-by-Step Solution

Answer

The rationalized form of \(\frac{13}{3+\sqrt{11}}\) is \(-19.5 + 6.5\sqrt{11}\).

Step 1: Identify the Conjugate

Identify the conjugate of the denominator \(3+\sqrt{11}\). The conjugate is found by changing the sign between the two terms. So the conjugate of \(3+\sqrt{11}\) is \(3-\sqrt{11}\).

Step 2: Multiply by the Conjugate

Multiply both the numerator and denominator of the fraction by the conjugate. This is like multiplying by 1, so it does not change the value of the fraction. In this case, we multiply by \(\frac{3-\sqrt{11}}{3-\sqrt{11}}\). Doing so results in:\[\frac{13}{3+\sqrt{11}} \times \frac{3-\sqrt{11}}{3-\sqrt{11}} = \frac{13\cdot (3-\sqrt{11})}{(3+\sqrt{11}) \cdot (3-\sqrt{11})}\]

Step 3: Distribute in the Numerator and use Difference of Squares in the Denominator

Distribute the 13 in the numerator and apply the rule of difference of squares in the denominator, which states that for any two terms \(a\) and \(b\), \((a+b)(a-b) = a^2 - b^2\).This leads to:\[\frac{39-13\sqrt{11}}{3^2 - \sqrt{11}^2} = \frac{39-13\sqrt{11}}{9-11} = \frac{39-13\sqrt{11}}{-2}\]

Step 4: Simplify

Simplify the fraction by dividing each term in the numerator by -2:\[\frac{39-13\sqrt{11}}{-2} = -19.5 + 6.5\sqrt{11}\]