What is the following quotient?
[tex]$\frac{9+\sqrt{2}}{4-\sqrt{7}}$[/tex]
Answer (2)
To simplify 4 − sqrt 7 9 + sqrt 2 , we multiply by the conjugate of the denominator, resulting in the simplified expression 9 36 + 9 sqrt 7 + 4 sqrt 2 + sqrt 14 . This method of rationalizing the denominator helps eliminate the square root from the denominator. Thus, the final answer is 9 36 + 9 sqrt 7 + 4 sqrt 2 + sqrt 14 .
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Multiply the numerator and denominator by the conjugate of the denominator.
Expand the numerator: ( 9 + 2 ) ( 4 + 7 ) = 36 + 9 7 + 4 2 + 14 .
Expand the denominator: ( 4 − 7 ) ( 4 + 7 ) = 9 .
The simplified expression is 9 36 + 9 7 + 4 2 + 14 .
Explanation
Understanding the Problem We are given the expression 4 − 7 9 + 2 and a list of possible values. Our goal is to determine which of the given values is equal to the given expression.
Rationalizing the Denominator To simplify the 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 4 − 7 is 4 + 7 .
Multiplying by the Conjugate Multiply the numerator and denominator by the conjugate: 4 − 7 9 + 2 × 4 + 7 4 + 7 = ( 4 − 7 ) ( 4 + 7 ) ( 9 + 2 ) ( 4 + 7 )
Expanding the Numerator Now, expand the numerator: ( 9 + 2 ) ( 4 + 7 ) = 9 ( 4 ) + 9 ( 7 ) + 4 ( 2 ) + 2 7 = 36 + 9 7 + 4 2 + 14
Expanding the Denominator Expand the denominator: ( 4 − 7 ) ( 4 + 7 ) = 4 2 − ( 7 ) 2 = 16 − 7 = 9
Simplifying the Expression So, the expression becomes: 9 36 + 9 7 + 4 2 + 14
Final Answer Comparing this result with the given options, we find that the correct answer is: 9 36 + 9 7 + 4 2 + 14
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 complex expressions involving irrational numbers. For example, when analyzing AC circuits, you might encounter expressions with complex numbers in the denominator. Rationalizing the denominator helps to separate the real and imaginary parts, making the calculations easier and more intuitive. This technique is also useful in optics when dealing with refractive indices.
