Question

Rationalize the denominator of each expression. Assume that all variables are positive when they appear. $$\frac{-3}{\sqrt{5}+4}$$

Short answer

\( \frac{3\sqrt{5} - 12}{11} \)

Step-by-step solution

- Identify the Conjugate

To rationalize the denominator, we need to use the conjugate of the denominator. The conjugate of \ \( \sqrt{5} + 4 \ \) is \ \( \sqrt{5} - 4 \ \).

- Multiply the Numerator and Denominator by the Conjugate

Multiply both the numerator and denominator by \ \( \sqrt{5} - 4 \ \): \[ \frac{-3}{\sqrt{5} + 4} \cdot \frac{\sqrt{5} - 4}{\sqrt{5} - 4} \]

- Apply the Distributive Property

Distribute the terms in the numerator and denominator: \[ \frac{-3(\sqrt{5} - 4)}{(\sqrt{5} + 4)(\sqrt{5} - 4)} \]

- Simplify the Denominator Using the Difference of Squares

The denominator will simplify using the difference of squares formula \( (a + b)(a - b) = a^2 - b^2 \): \[ (\sqrt{5})^2 - 4^2 = 5 - 16 = -11 \]

- Simplify the Numerator

Simplify the top part: \[ -3(\sqrt{5} - 4) = -3\sqrt{5} + 12 \]

- Combine the Results

Combine the simplified numerator and denominator: \[ \frac{-3\sqrt{5} + 12}{-11} = \frac{3\sqrt{5} - 12}{11} \]

Key concepts

Conjugate

When we talk about the conjugate in math, we're dealing with a special pair of binomials. To rationalize a denominator with a radical, we use the conjugate of the radical expression. For instance, if you have \(\sqrt{5} + 4\), the conjugate is \(\sqrt{5} - 4\). Notice that only the sign between the terms changes; the values themselves remain the same. The purpose of using the conjugate is to create a situation where the product of these terms results in a rational number (an integer in most cases), making the fraction easier to handle.

Difference of Squares

The difference of squares is a specific algebraic identity used in rationalizing denominators. It states that \( (a + b)(a - b) = a^2 - b^2\). When you multiply a binomial by its conjugate, you utilize this identity. For example, in the problem \( \frac{-3}{\sqrt{5} + 4} \), we multiply both the numerator and denominator by \(\sqrt{5} - 4\). The denominator becomes \( (\sqrt{5} + 4)(\sqrt{5} - 4) \). Applying the difference of squares, we get: \[ (\sqrt{5})^2 - 4^2 = 5 - 16 = -11 \]. This simplifies the denominator to a rational number.

Distributive Property

The distributive property is a fundamental algebraic property that allows us to multiply a single term and two or more terms inside a set of parentheses. According to this property, \( a(b + c) = ab + ac \). In the context of the given exercise, we see it applied as follows: \( -3(\sqrt{5} - 4) \), resulting in \[ -3 \cdot \sqrt{5} + -3 \cdot -4 = -3\sqrt{5} + 12 \]. This step ensures that every term is multiplied correctly, leading to a fully simplified numerator. Combining this with the newly rationalized denominator completes the process: \ \frac{3\sqrt{5} - 12}{11} \. Using the distributive property helps distribute the multiplication correctly across all terms.