To rationalize the denominator of [tex]$\frac{5-\sqrt{7}}{9-\sqrt{14}}$[/tex], you should multiply the expression by which fraction?
[tex]$\frac{5+\sqrt{7}}{8-\sqrt{14}}$[/tex]
[tex]$\frac{8-\sqrt{14}}{8-\sqrt{14}}$[/tex]
[tex]$\frac{8+\sqrt{14}}{8+\sqrt{14}}$[/tex]
[tex]$\frac{\sqrt{14}}{\sqrt{14}}$[/tex]

Question

Anonymous · Answer

To rationalize the denominator of 9 − s q r t 14 5 − s q r t 7 ​ , multiply by 9 + s q r t 14 9 + s q r t 14 ​ . This process uses the conjugate of the denominator to eliminate the square root. Therefore, the chosen option is 9 + s q r t 14 9 + s q r t 14 ​ . 
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GinnyAnswer · Answer

Identify the denominator: 9 − 14 ​ . 
 Find the conjugate of the denominator: 9 + 14 ​ . 
 Multiply the expression by the fraction formed by the conjugate: 9 + 14 ​ 9 + 14 ​ ​ . 
 The fraction to multiply by is 9 + 14 ​ 9 + 14 ​ ​ ​ . 
 
 Explanation 
 
 Understanding the Problem The goal is to rationalize the denominator of the fraction 9 − 14 ​ 5 − 7 ​ ​ . Rationalizing the denominator means we want to get rid of the square root in the denominator. To do this, we multiply the numerator and denominator by the conjugate of the denominator. 
 
 Finding the Conjugate The denominator is 9 − 14 ​ . The conjugate of 9 − 14 ​ is 9 + 14 ​ . Therefore, we need to multiply the numerator and denominator by 9 + 14 ​ . This is equivalent to multiplying the entire expression by the fraction 9 + 14 ​ 9 + 14 ​ ​ . 
 
 Conclusion Therefore, to rationalize the denominator of 9 − 14 ​ 5 − 7 ​ ​ , we should multiply the expression by 9 + 14 ​ 9 + 14 ​ ​ .

Examples 
 Rationalizing the denominator is a technique used in various mathematical and scientific contexts to simplify expressions and make them easier to work with. For example, in physics, when dealing with impedance in electrical circuits, you might encounter complex numbers in the denominator. Rationalizing the denominator helps in expressing the impedance in a standard form, making it easier to calculate and analyze circuit behavior. Similarly, in chemistry, when calculating concentrations or equilibrium constants, rationalizing denominators can simplify calculations and provide clearer results.

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