Question

Rationalize the numerator of each expression. Assume that all variables are positive when they appear. $$\frac{\sqrt{x}-2}{x-4} x \neq 4$$

Short answer

The rationalized form is \(\frac{1}{\text{\textbackslash sqrt\textbraceleft x\textbraceright}+2}\).

Step-by-step solution

Identify the Conjugate

Identify the conjugate of the numerator. The original numerator is \(\frac{\text{\textbackslash sqrt\textbraceleft x\textbraceright}-2}{x-4}\). The conjugate of \(\text{\textbackslash sqrt\textbraceleft x\textbraceright}-2\) is \(\text{\textbackslash sqrt\textbraceleft x\textbraceright}+2\).

Multiply by the Conjugate

Multiply both the numerator and the denominator by the conjugate \(\text{\textbackslash sqrt\textbraceleft x\textbraceright}+2\). This gives: \(\frac{(\text{\textbackslash sqrt\textbraceleft x\textbraceright}-2)(\text{\textbackslash sqrt\textbraceleft x\textbraceright}+2)}{(x-4)(\text{\textbackslash sqrt\textbraceleft x\textbraceright}+2)}\).

Apply the Difference of Squares

Use the difference of squares formula \(a^2 - b^2 = (a-b)(a+b)\) to simplify the numerator: \((\text{\textbackslash sqrt\textbraceleft x\textbraceright}-2)(\text{\textbackslash sqrt\textbraceleft x\textbraceright}+2) = x-4\). The expression now becomes \(\frac{x-4}{(x-4)(\text{\textbackslash sqrt\textbraceleft x\textbraceright}+2)}\).

Simplify the Fraction

Cancel the common factor in the numerator and the denominator: \(\frac{x-4}{(x-4)(\text{\textbackslash sqrt\textbraceleft x\textbraceright}+2)} = \frac{1}{\text{\textbackslash sqrt\textbraceleft x\textbraceright}+2}\).

Key concepts

conjugate of a radical expression

When rationalizing the numerator of a fraction, the first step is often to use the conjugate of the numerator. The conjugate of a radical expression involves changing the sign between two terms. For example, the conjugate of \(\sqrt{x} - 2\) is \(\sqrt{x} + 2\). This technique is very useful because it allows us to remove the square root in the numerator when multiplied properly.

Multiplying by the conjugate helps in making the numerator a rational expression, which can be simplified further using algebraic properties.

difference of squares

The difference of squares is a mathematical formula used to simplify expressions that are in the format of \(a^2 - b^2 = (a - b)(a + b)\). This identity is particularly useful in our problem when we have \(\sqrt{x} - 2\) and \(\sqrt{x} + 2\).

Applying the difference of squares to our example, we multiply \(\sqrt{x} - 2\) and \(\sqrt{x} + 2\): \[ (\sqrt{x} - 2)(\sqrt{x} + 2) = x - 4 \] This simplifies \(\sqrt{x} - 2\) times its conjugate into a term without a square root, making it easier to work with further in rationalizing the numerator.

simplify fractions

Simplifying fractions involves reducing the fraction to its simplest form by canceling common factors in the numerator and the denominator. After applying the difference of squares in our problem, we have the fraction: \[ \frac{x-4}{(x-4)(\sqrt{x}+2)} \]

Here, \(x - 4 \) is a common factor in both the numerator and the denominator. We cancel out \(x - 4\) from both, leaving us with: \[ \frac{1}{\sqrt{x} + 2} \] This is the simplest form of the fraction, completing the process of rationalizing the numerator. Simplification like this is crucial to make expressions easier to understand and work with in further mathematical calculations.