What is the following quotient?
$\frac{5}{\sqrt{11}-\sqrt{3}}$
Answer (2)
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 ) 2 − ( 3 ) 2 = 11 − 3 = 8 .
The expression becomes: 8 5 ( 11 + 3 ) .
The simplified expression is: 8 5 11 + 5 3 .
Explanation
Understanding the Problem We are asked to simplify the expression 11 − 3 5 . This involves rationalizing the denominator.
Rationalizing the Denominator To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator, which is 11 + 3 . This gives us: 11 − 3 5 × 11 + 3 11 + 3 = ( 11 − 3 ) ( 11 + 3 ) 5 ( 11 + 3 ) .
Simplifying the Denominator Now, 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 Simplified Expression So, the expression becomes: 8 5 ( 11 + 3 ) = 8 5 11 + 5 3 .
Identifying the Correct Option Comparing this with the given options, we see that the correct answer is 8 5 11 + 5 3 .
Examples
Rationalizing the denominator is a technique used to eliminate radicals from the denominator of a fraction. This is useful in various fields, such as physics and engineering, where simplified expressions are easier to work with. For example, when calculating impedance in electrical circuits, you might encounter complex numbers with radicals in the denominator. Rationalizing the denominator makes it easier to perform further calculations and comparisons.
To simplify 11 − 3 5 , we multiply by the conjugate to get 8 5 ( 11 + 3 ) , simplifying to 8 5 11 + 5 3 . This process rationalizes the denominator, making the expression easier to work with. So the final answer is 8 5 11 + 5 3 .
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