What is the following quotient?
[tex]$\frac{1}{1+\sqrt{3}}$[/tex]
[tex]$\frac{\sqrt{3}}{4}$[/tex]
[tex]$\frac{1+\sqrt{3}}{4}$[/tex]
[tex]$\frac{1-\sqrt{3}}{4}$[/tex]
[tex]$\frac{-1+\sqrt{3}}{2}$[/tex]
Answer (2)
The expression \frac{1}{1+\text{\sqrt{3}}} simplifies to \frac{-1 + \text{\sqrt{3}}}{2} . Thus, the correct answer choice is \frac{-1+\text{\sqrt{3}}}{2} .
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∙ Rationalize the denominator by multiplying the numerator and denominator by the conjugate of 1 + 3 , which is 1 − 3 .
∙ Simplify the denominator using the difference of squares: ( 1 + 3 ) ( 1 − 3 ) = 1 − 3 = − 2 .
∙ The expression becomes − 2 1 − 3 .
∙ Multiply the numerator and denominator by -1 to obtain the final simplified form: 2 − 1 + 3 .
Explanation
Understanding the Problem We are asked to simplify the expression 1 + 3 1 and determine which of the provided options is equivalent.
Rationalizing the Denominator To simplify the given expression, we need to rationalize the denominator. This means we want to get rid of the square root in the denominator. We can do this by multiplying both the numerator and the denominator by the conjugate of the denominator. The conjugate of 1 + 3 is 1 − 3 .
Multiplying by the Conjugate Multiply the numerator and denominator by the conjugate: 1 + 3 1 ⋅ 1 − 3 1 − 3 = ( 1 + 3 ) ( 1 − 3 ) 1 − 3
Simplifying the Denominator Now, we simplify the denominator using the difference of squares formula, which states that ( a + b ) ( a − b ) = a 2 − b 2 . In our case, a = 1 and b = 3 . So, we have: ( 1 + 3 ) ( 1 − 3 ) = 1 2 − ( 3 ) 2 = 1 − 3 = − 2
Substituting Back Substitute the simplified denominator back into the expression: − 2 1 − 3
Final Simplification To make the expression look more like one of the answer choices, we can multiply both the numerator and the denominator by -1: − 2 1 − 3 ⋅ − 1 − 1 = 2 − 1 + 3
Conclusion Therefore, the simplified expression is 2 − 1 + 3 , which matches one of the provided options.
Examples
Rationalizing the denominator is a technique used in various fields, such as electrical engineering when dealing with impedance calculations or in physics when simplifying expressions involving complex numbers. For example, when calculating the equivalent impedance of a circuit, you might encounter an expression with a complex number in the denominator. Rationalizing the denominator helps to express the impedance in a standard form, making it easier to analyze and design the circuit. Suppose you have an impedance Z = 2 + j 3 1 , where j is the imaginary unit. To rationalize the denominator, you multiply both the numerator and denominator by the conjugate of the denominator, which is 2 − j 3 . This gives you Z = ( 2 + j 3 ) ( 2 − j 3 ) 2 − j 3 = 4 + 9 2 − j 3 = 13 2 − j 3 . This form is easier to work with for further calculations.
