What is the following quotient?
[tex]$\frac{9+\sqrt{2}}{4-\sqrt{7}}$[/tex]
[tex]$\frac{9 \sqrt{7}+\sqrt{14}}{-3}$[/tex]
[tex]$\frac{36-9 \sqrt{7}+4 \sqrt{2}-\sqrt{14}}{9}$[/tex]
[tex]$\frac{36+9 \sqrt{7}+4 \sqrt{2}+\sqrt{14}}{9}$[/tex]
[tex]$\frac{79}{9}$[/tex]
Answer (2)
To simplify \frac{9+\text{\sqrt{2}}}{4-\text{\sqrt{7}}} , we rationalize the denominator by multiplying by its conjugate. This results in 9 36 + 9 7 + 4 2 + 14 , which matches one of the provided options. Thus, the correct answer is 9 36 + 9 7 + 4 2 + 14 .
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Multiply the numerator and denominator by the conjugate of the denominator: 4 + 7 4 + 7 .
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
Problem Analysis 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 4 + 7 : 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
Simplified Expression Therefore, 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 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 makes it easier to perform further calculations and understand the physical meaning of the expression.
