What is the following quotient?
$\frac{5}{\sqrt{11}-\sqrt{3}}$
Answer (2)
To simplify 11 − 3 5 , we multiply by the conjugate 11 + 3 to eliminate the square root in the denominator. This gives the final result of 8 5 11 + 5 3 .
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Multiply the numerator and denominator by the conjugate of the denominator: 11 − 3 5 × 11 + 3 11 + 3 .
Simplify the denominator using the difference of squares: ( 11 − 3 ) ( 11 + 3 ) = 11 − 3 = 8 .
The expression simplifies to 8 5 ( 11 + 3 ) .
Distribute the 5 in the numerator to get the final answer: 8 5 11 + 5 3 .
Explanation
Understanding the Problem We are asked to simplify the expression 11 − 3 5 . To do this, we will rationalize the denominator by multiplying both the numerator and the denominator by the conjugate of the denominator.
Rationalizing the Denominator The conjugate of 11 − 3 is 11 + 3 . So, we multiply the numerator and denominator by this conjugate: 11 − 3 5 × 11 + 3 11 + 3 .
Multiplying by the Conjugate Now, we multiply the numerators and denominators: ( 11 − 3 ) ( 11 + 3 ) 5 ( 11 + 3 ) .
Simplifying the Denominator We simplify the denominator using the difference of squares formula, ( a − b ) ( a + b ) = a 2 − b 2 : ( 11 − 3 ) ( 11 + 3 ) = ( 11 ) 2 − ( 3 ) 2 = 11 − 3 = 8 .
Final Simplification So, the expression becomes: 8 5 ( 11 + 3 ) = 8 5 11 + 5 3 .
Choosing the Correct Answer Comparing this result with the given options, we find that it matches the second option.
Final Answer Therefore, the simplified expression is 8 5 11 + 5 3 .
Examples
Rationalizing the denominator is a useful technique in various fields, such as physics and engineering, where simplified expressions are crucial for calculations. For example, when dealing with impedance in electrical circuits or calculating forces in mechanics, simplifying expressions involving radicals can make the problem easier to solve. Consider an electrical circuit where the impedance is given by a complex number with a radical in the denominator. Rationalizing the denominator allows engineers to easily determine the real and imaginary parts of the impedance, which are essential for circuit analysis and design. Suppose the impedance Z is given by Z = 5 − 2 10 . By rationalizing the denominator, we get Z = 5 − 2 10 ( 5 + 2 ) = 3 10 ( 5 + 2 ) . This simplified form makes it easier to work with the impedance in further calculations.
