Problem 363 - Elementary Algebra

Question

In the following exercises, simplify by rationalizing the denominator. (a) \(\frac{3}{3+\sqrt{11}}\) (b) \(\frac{8}{1-\sqrt{5}}\)

Step-by-Step Solution

Answer

(a) -\frac{9}{2} + \frac{3\sqrt{11}}{2}, (b) -2 - 2\sqrt{5}.

Step 1: Identify the denominator

For each fraction, observe the denominator which contains a radical. For (a), the denominator is \(3 + \sqrt{11}\) and for (b), the denominator is \(1 - \sqrt{5}\).

Step 2: Multiply by the conjugate

To rationalize the denominator, multiply both the numerator and the denominator of each fraction by the conjugate of the denominator. For (a), the conjugate of \(3 + \sqrt{11}\) is \(3 - \sqrt{11}\). For (b), the conjugate of \(1 - \sqrt{5}\) is \(1 + \sqrt{5}\). Thus, multiply: \[ \frac{3}{3+\sqrt{11}} \times \frac{3-\sqrt{11}}{3-\sqrt{11}} \] and \[ \frac{8}{1-\sqrt{5}} \times \frac{1+\sqrt{5}}{1+\sqrt{5}} \]

Step 3: Simplify the numerator

For each fraction, carry out the multiplication in the numerator: (a) \[3 \times (3 - \sqrt{11}) = 3(3) - 3(\sqrt{11}) = 9 - 3\sqrt{11}\] (b) \[8 \times (1 + \sqrt{5}) = 8(1) + 8(\sqrt{5}) = 8 + 8\sqrt{5}\]

Step 4: Simplify the denominator

Use the difference of squares formula to simplify the denominator in each fraction: (a) \[ (3 + \sqrt{11})(3 - \sqrt{11}) = 3^2 - (\sqrt{11})^2 = 9 - 11 = -2\] (b) \[ (1 - \sqrt{5})(1 + \sqrt{5}) = 1^2 - (\sqrt{5})^2 = 1 - 5 = -4\]

Step 5: Combine and simplify the fraction

Combine the simplified numerators and denominators: (a) \[ \frac{9 - 3\sqrt{11}}{-2} = -\frac{9}{2} + \frac{3 \sqrt{11}}{2} = -\frac{9}{2} + \frac{3\sqrt{11}}{2}\] (b) \[ \frac{8 + 8\sqrt{5}}{-4} = -2 - 2\sqrt{5}\]

Key Concepts

Conjugate

When we rationalize the denominator, we often need to use the conjugate. The conjugate of a binomial (a two-term expression) is the same two terms with the opposite sign in between. For example, the conjugate of \(3 + \sqrt{11}\) is \(3 - \sqrt{11}\). This process helps to eliminate the radical in the denominator.

To rationalize the denominator, multiply both the numerator and denominator by the conjugate of the denominator. This technique leverages the difference of squares formula to simplify the expression. The steps involve identifying the conjugate and multiplying the entire fraction by \frac{conjugate}{conjugate}\.

It's very important to apply the same operation (multiplying with the conjugate) to the numerator as well, maintaining the value of the original fraction.

Simplifying Fractions

Simplifying fractions requires multiplying and reducing the numerator and the denominator correctly. Once you've multiplied the fraction by the conjugate, you'll need to carry out the multiplication within both the numerator and the denominator.

For instance, let's consider the fraction \frac{3}{3 + \sqrt{11}}.\ After multiplying by the conjugate \frac{3 - \sqrt{11}}{3 - \sqrt{11}} we need to simplify each part.

Numerators simplify by distributing the terms. For \(3 \times (3 - \sqrt{11})\):
\[3(3) - 3(\sqrt{11}) = 9 - 3\sqrt{11}\]

Numerator for \frac{8}{1 - \sqrt{5}}\ becomes:
\[8(1 + \sqrt{5}) = 8 + 8\sqrt{5}\]

Be careful to manage negative and positive sign changes correctly and reduce the simplifications where possible.

Difference of Squares

The difference of squares formula is \(a^2 - b^2 = (a + b)(a - b)\). This is crucial in the process of rationalizing and simplifying expressions that contain radicals.

In the conjugate multiplication step, the product of conjugates results in a difference of squares: orthen with applying the formula.

For example:\( (3 + \sqrt{11})(3 - \sqrt{11}) = 3^2 - (\sqrt{11})^2 \) simplifying to \(9 - 11 = -2\).

Similarly, when we multiply \ (1 - \sqrt{5})(1 + \sqrt{5}) \, it simplifies as:
\(1^2 - (\sqrt{5})^2 = 1 - 5 = -4.\)

Using the difference of squares allows us to convert our expression into a rational number, removing the radical from the denominator.