Rationalize the denominator and simplify: [tex]$\frac{x}{x+\sqrt{3}}$[/tex]
Answer (2)
Multiply the numerator and denominator by the conjugate of the denominator: 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 we want to eliminate the square root from the denominator.
Multiplying by the Conjugate To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of x + 3 is x − 3 . So, we multiply the expression by x − 3 x − 3 : x + 3 x × x − 3 x − 3
Expanding the Expression Now, we multiply out the numerator and the denominator. The numerator becomes: x ( x − 3 ) = x 2 − x 3 The denominator becomes: ( x + 3 ) ( x − 3 ) = x 2 − ( 3 ) 2 = x 2 − 3
Rationalized Expression So, the expression becomes: x 2 − 3 x 2 − x 3
Final Answer The expression x 2 − 3 x 2 − x 3 is now in its simplest form, with a rationalized denominator.
Examples
Rationalizing the denominator is a technique used in various fields, such as physics and engineering, when dealing with complex numbers or expressions in order to simplify calculations and make them easier to work with. For example, when calculating impedance in electrical circuits or dealing with wave functions in quantum mechanics, rationalizing denominators can help in obtaining more manageable and interpretable results. Consider a scenario where you are designing a bridge and need to calculate the forces acting on a support beam. If your calculations involve expressions with irrational numbers in the denominator, rationalizing them can simplify the process and lead to more accurate results.
To rationalize the denominator of x + 3 x , multiply by the conjugate x − 3 . The final expression after simplification is x 2 − 3 x 2 − x 3 .
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