Question

Rationalize the numerator of each expression. Assume that all variables are positive when they appear. $$\frac{\sqrt{11}+1}{2}$$

Short answer

\[ \frac{10}{2\sqrt{11} - 2} \]

Step-by-step solution

- Identify the Conjugate

To rationalize the numerator, identify the conjugate of the numerator. The numerator is \( \sqrt{11} + 1 \). The conjugate of \( \sqrt{11} + 1 \) is \( \sqrt{11} - 1 \).

- Multiply by the Conjugate

Multiply the numerator and the denominator by the conjugate of the numerator: \[ \frac{\sqrt{11} + 1}{2} \times \frac{\sqrt{11} - 1}{\sqrt{11} - 1} \]

- Apply the Difference of Squares

Simplify the multiplication in the numerator using the difference of squares formula \( (a+b)(a-b) = a^2 - b^2 \): \[ (\sqrt{11} + 1)(\sqrt{11} - 1) = (\sqrt{11})^2 - (1)^2 = 11 - 1 = 10 \]

- Simplify the Denominator

Multiply the denominators: \[ 2 \times (\sqrt{11} - 1) = 2\sqrt{11} - 2 \]

- Combine the Results

Put the simplified numerator and the denominator together: \[ \frac{10}{2(\sqrt{11} - 1)} = \frac{10}{2\sqrt{11} - 2} \]

Key concepts

conjugate

When you want to rationalize the numerator, working with the conjugate is key.
The conjugate of a given expression is found by changing the sign between two terms. For example, if the original term is \(a + b\), the conjugate would be \(a - b\).

This technique capitalizes on creating a simpler form of the expression. For instance, with the numerator \( \sqrt{11} + 1 \), its conjugate is \( \sqrt{11} - 1 \).

This leads us to the term 'difference of squares,' which we'll explore next.

difference of squares

The concept of difference of squares helps simplify expressions when we multiply conjugates.
The formula for the difference of squares is \( (a + b)(a - b) = a^2 - b^2 \).

Let's use our example's numerator to illustrate. We have \( (\sqrt{11} + 1)(\sqrt{11} - 1) \).
According to the formula:

• \(a = \sqrt{11}\)
• \(b = 1\)

Plugging in these values, we get: \( (\sqrt{11})^2 - (1)^2 = 11 - 1 = 10 \).

This simplifies the fraction significantly. Finally, let's talk about simplifying the entire expression.

simplifying expressions

Simplifying expressions involves making the numerator and the denominator as simple as possible.
Once you've rationalized the numerator using the conjugate and difference of squares, the fraction transforms.

For our example, the rationalized form in the numerator is 10. Next, simplify the denominator by straightforward multiplication: \( 2(\sqrt{11} - 1) = 2\sqrt{11} - 2 \).

Now, substitute the simplified numerator and the denominator back: \ \frac{10}{2(\sqrt{11} - 1)} = \frac{10}{2\sqrt{11} - 2} \ which simplifies you the expression correctly and fully.

Practicing these steps aids students in confidently handling similar problems, ensuring they grasp rationalizing numerators fully and completely.