What is the following quotient?
$\frac{\sqrt{6}+\sqrt{11}}{\sqrt{5}+\sqrt{3}}$
Answer (2)
To simplify 5 + 3 6 + 11 , we rationalize the denominator and derive the expression 2 30 − 3 2 + 55 − 33 . This is achieved by multiplying by the conjugate of the denominator and simplifying. The final result is 2 30 − 3 2 + 55 − 33 .
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Rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator.
Simplify the denominator: ( 5 + 3 ) ( 5 − 3 ) = 5 − 3 = 2 .
Expand and simplify the numerator: ( 6 + 11 ) ( 5 − 3 ) = 30 − 3 2 + 55 − 33 .
The simplified expression is 2 30 − 3 2 + 55 − 33 , which matches the second option: 2 30 − 3 2 + 55 − 33 .
Explanation
Understanding the Problem We are given the expression 5 + 3 6 + 11 and five possible values. We want to determine which of the five values is equal to the given expression.
Rationalizing the Denominator To find the equivalent expression, we can rationalize the denominator of the given expression by multiplying both the numerator and the denominator by the conjugate of the denominator. The conjugate of 5 + 3 is 5 − 3 .
Multiplying by the Conjugate Multiply the numerator and denominator by the conjugate: 5 + 3 6 + 11 × 5 − 3 5 − 3 = ( 5 + 3 ) ( 5 − 3 ) ( 6 + 11 ) ( 5 − 3 )
Simplifying the Denominator First, simplify the denominator: ( 5 + 3 ) ( 5 − 3 ) = ( 5 ) 2 − ( 3 ) 2 = 5 − 3 = 2
Expanding the Numerator Next, expand the numerator: ( 6 + 11 ) ( 5 − 3 ) = 6 5 − 6 3 + 11 5 − 11 3 = 30 − 18 + 55 − 33
Simplifying the Numerator Terms Simplify the terms in the numerator: 30 − 18 + 55 − 33 = 30 − 9 × 2 + 55 − 33 = 30 − 3 2 + 55 − 33
Combining the Results Therefore, the expression simplifies to: 2 30 − 3 2 + 55 − 33
Identifying the Correct Answer Comparing this result with the given options, we find that it matches the second option. 2 30 − 3 2 + 55 − 33
Examples
Rationalizing the denominator is a useful technique in various fields, such as electrical engineering when dealing with impedance calculations or in physics when simplifying complex expressions involving radicals. For example, when calculating the equivalent resistance in an AC circuit, you might encounter complex numbers with radicals in the denominator. Rationalizing the denominator helps in simplifying these expressions, making them easier to work with and interpret. This technique ensures that the final result is expressed in a standard form, which is crucial for accurate calculations and practical applications.
