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 by multiplying by its conjugate, which leads to the final expression of 2 30 − 3 2 + 55 − 33 .
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Rationalize the denominator by multiplying by the conjugate.
Simplify the denominator: ( 5 + 3 ) ( 5 − 3 ) = 2 .
Expand the numerator: ( 6 + 11 ) ( 5 − 3 ) = 30 − 3 2 + 55 − 33 .
The simplified expression is 2 30 − 3 2 + 55 − 33 , which matches option 2. Therefore, the answer is 2 30 − 3 2 + 55 − 33 .
Explanation
Understanding the Problem We are given the expression 5 + 3 6 + 11 and four possible values. Our objective is to determine which of the four values is equal to the given expression.
Rationalizing the Denominator To simplify the expression, we will rationalize the denominator 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 the denominator by the conjugate: 5 + 3 6 + 11 × 5 − 3 5 − 3 = ( 5 + 3 ) ( 5 − 3 ) ( 6 + 11 ) ( 5 − 3 )
Simplifying the Denominator First, let's simplify the denominator: ( 5 + 3 ) ( 5 − 3 ) = ( 5 ) 2 − ( 3 ) 2 = 5 − 3 = 2
Expanding the Numerator Now, let's expand the numerator: ( 6 + 11 ) ( 5 − 3 ) = 6 5 − 6 3 + 11 5 − 11 3 = 30 − 18 + 55 − 33 Since 18 = 9 × 2 = 3 2 , the numerator becomes: 30 − 3 2 + 55 − 33
Simplifying the Expression So, the entire expression simplifies to: 2 30 − 3 2 + 55 − 33
Finding 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 total impedance in an AC circuit, you might encounter complex numbers with radicals in the denominator. Rationalizing the denominator helps to express the impedance in a standard form, making it easier to analyze the circuit's behavior. This technique ensures that the imaginary part of the impedance is clearly separated, which is crucial for designing and troubleshooting electrical systems.
