How Do You Rationalize the Denominator to Solve for a+b?

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Homework Help Overview

The problem involves rationalizing the denominator of a square root expression to express it in the form \(a + b\sqrt{6}\). The original poster presents a mathematical expression involving square roots and seeks to determine the values of \(a\) and \(b\).

Discussion Character

  • Exploratory, Mathematical reasoning, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the process of rationalizing the denominator and express confusion about converting the resulting expression into the desired form. There are attempts to manipulate the expression and questions about the structure of the numerator.

Discussion Status

Some participants have offered guidance on squaring both sides of the equation and rationalizing the denominator. There is ongoing exploration of how to express the result in the required format, with various interpretations of the steps involved.

Contextual Notes

There is a noted uncertainty about the correct manipulation of the expression and the assumptions regarding the form of the numerator. Participants are also considering the implications of their calculations on the values of \(a\) and \(b\).

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Homework Statement



If ##\sqrt{\frac{10+4\sqrt{6}}{10-4\sqrt{6}}}=a+b\sqrt{6}##, then a+b is ?

A) 8
B) 7
C) 6
D) 5
E) 4

The Attempt at a Solution


[/B]
This is my attempt:
##\sqrt{\frac{10+4\sqrt{6}}{10-4\sqrt{6}}}=\sqrt{\frac{10+4\sqrt{6}*(10+4\sqrt{6})}{10-4\sqrt{6}*(10+4\sqrt{6})}}=\sqrt{\frac{196+80\sqrt{6}}{4}}=\sqrt{49+20\sqrt{6}}##

Then, I got stuck.. I have no idea how to convert to form a+b√6
Please help..
 
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terryds said:

Homework Statement



If ##\sqrt{\frac{10+4\sqrt{6}}{10-4\sqrt{6}}}=a+b\sqrt{6}##, then a+b is ?

A) 8
B) 7
C) 6
D) 5
E) 4

The Attempt at a Solution


[/B]
This is my attempt:
##\sqrt{\frac{10+4\sqrt{6}}{10-4\sqrt{6}}}=\sqrt{\frac{10+4\sqrt{6}*(10+4\sqrt{6})}{10-4\sqrt{6}*(10+4\sqrt{6})}}=\sqrt{\frac{196+80\sqrt{6}}{4}}=\sqrt{49+20\sqrt{6}}##

Then, I got stuck.. I have no idea how to convert to form a+b√6
Please help..
Square both sides of your original equation and then rationalize the denominator on the left side.
You should get something like m + n√(6) on one side and r + s√(6) on the other side. You can equate m with r and n with s to get two equations involving a and b.
 
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terryds said:
This is my attempt:
##\sqrt{\frac{10+4\sqrt{6}}{10-4\sqrt{6}}}=\sqrt{\frac{10+4\sqrt{6}*(10+4\sqrt{6})}{10-4\sqrt{6}*(10+4\sqrt{6})}}=\sqrt{\frac{196+80\sqrt{6}}{4}}=\sqrt{49+20\sqrt{6}}##

.
You have to use parentheses. The correct form is ##\sqrt{\frac{10+4\sqrt{6}}{10-4\sqrt{6}}}=\sqrt{\frac{(10+4\sqrt{6})*(10+4\sqrt{6})}{(10-4\sqrt{6})*(10+4\sqrt{6})}}##
Is not the numerator the square of something? What is its square root?
 
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Mark44 said:

Square both sides of your original equation and then rationalize the denominator on the left side.
You should get something like m + n√(6) on one side and r + s√(6) on the other side. You can equate m with r and n with s to get two equations involving a and b.
Thanks.. a is 5 and b is 2 , so a+b is 7..
 

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