Rationalize the denominator and simplify: [tex]$\frac{x}{x+\sqrt{3}}$[/tex]
Answer (1)
Multiply the numerator and denominator by the conjugate of the denominator: x + 3 x × x − 3 x − 3 .
Expand the numerator: x ( x − 3 ) = x 2 − x 3 .
Expand the denominator: ( x + 3 ) ( x − 3 ) = x 2 − 3 .
The simplified expression is: x 2 − 3 x 2 − x 3 .
Explanation
Understanding the Problem We are given the expression x + 3 x . Our goal is to rationalize the denominator, which means eliminating the square root from the denominator. To do this, we multiply both the numerator and the denominator by the conjugate of the denominator.
Multiplying by the Conjugate The conjugate of x + 3 is x − 3 . So, we multiply both the numerator and the denominator by x − 3 : x + 3 x × x − 3 x − 3
Expanding the Expression Now, we multiply out the numerator and the denominator. u m er a t or = x ( x − 3 ) = x 2 − x 3 \denominator = ( x + 3 ) ( x − 3 ) = x 2 − ( 3 ) 2 = x 2 − 3
Rationalized and Simplified Expression So, the expression becomes: x 2 − 3 x 2 − x 3
Final Answer Therefore, the rationalized and simplified form of the given expression is x 2 − 3 x 2 − x 3 .
Examples
Rationalizing the denominator is a technique used in various mathematical and scientific contexts to simplify expressions and make them easier to work with. For example, in electrical engineering, when dealing with impedance calculations involving complex numbers, rationalizing the denominator helps in separating the real and imaginary parts, making it easier to analyze the circuit's behavior. Similarly, in optics, when calculating the reflection and transmission coefficients of light waves at an interface, rationalizing the denominator simplifies the expressions and allows for easier interpretation of the results.
