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Rationalizing the Denominator using ConjugatesRationalizing the Denominator using Conjugates
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Question 1 of 4
1. Question
Express the following with rational denominators:
`1/(2+sqrt2)`
Correct
Excellent!
Incorrect
Difference of Two Squares
`(a+b)(ab)=a^2b^2`The conjugate is where we change the sign in the middle of two terms like this:
For the answer to be in simplest form, the denominator should be a rational number.Multiply the numerator and the denominator by the conjugate of the denominator which is: `color(crimson)(2sqrt2)``1/(2+sqrt2)` `=` `1/(2+sqrt2) xx color(crimson)((2sqrt2)/(2sqrt2))` `=` `(2sqrt2)/((2+sqrt2)(2sqrt2))` Simplify using the difference of two squares `=` `(2sqrt2)/((2)^2(sqrt2)^2)` Simplify `=` `(2sqrt2)/(42)` `(2)^2 = 4` and `(sqrt(2))^2 = 2` `=` `(2sqrt2)/2` `(2sqrt2)/2` 
Question 2 of 4
2. Question
Express the following with rational denominators:
`1/(sqrt6sqrt5)`
Correct
Excellent!
Incorrect
Difference of Two Squares
`(a+b)(ab)=a^2b^2`The conjugate is where we change the sign in the middle of two terms like this:
For the answer to be in simplest form, the denominator should be a rational number.Multiply the numerator and the denominator by the conjugate of the denominator which is: `color(crimson)(sqrt6+sqrt5)``1/(sqrt6sqrt5)` `=` `1/(sqrt6sqrt5) xx color(crimson)((sqrt6+sqrt5)/(sqrt6+sqrt5))` `=` `(sqrt6+sqrt5)/((sqrt6sqrt5)(sqrt6+sqrt5))` Simplify using the difference of two squares `=` `(sqrt6+sqrt5)/((sqrt6)^2(sqrt5)^2)` Simplify square roots `=` `(sqrt6+sqrt5)/(65)` `(sqrt(36))^2 = 6` and `(sqrt(25))^2 = 5` `=` `(sqrt6+sqrt5)/1` `=` `sqrt6+sqrt5` `sqrt6+sqrt5` 
Question 3 of 4
3. Question
Express the following with rational denominators:
`4/(4sqrt2)`
Correct
Excellent!
Incorrect
Difference of Two Squares
`(a+b)(ab)=a^2b^2`The conjugate is where we change the sign in the middle of two terms like this:
For the answer to be in simplest form, the denominator should be a rational number.Multiply the numerator and the denominator by the conjugate of the denominator which is: `color(crimson)(4+sqrt2)``4/(4sqrt2)` `=` `4/(4sqrt2) xx color(crimson)((4+sqrt2)/(4+sqrt2))` `=` `(4 xx 4+4sqrt2)/((4sqrt2)(4+sqrt2))` Simplify using the difference of two squares `=` `(16+4sqrt2)/(4^2(sqrt2)^2)` Simplify square roots `=` `(16+4sqrt2)/162` `4^2 = 16` and `(sqrt(2))^2 = 2` `=` `(16 color(crimson)(:2)+4sqrt2 color(crimson)(:2))/(14 color(crimson)(:2))` Simplify by dividing throughout by `2` `=` `(8+2sqrt2)/7` `(8+2sqrt2)/7` 
Question 4 of 4
4. Question
Express the following with rational denominators:
`12/(sqrt7sqrt3)`
Correct
Excellent!
Incorrect
Difference of Two Squares
`(a+b)(ab)=a^2b^2`The conjugate is where we change the sign in the middle of two terms like this:
For the answer to be in simplest form, the denominator should be a rational number.Multiply the numerator and the denominator by the conjugate of the denominator which is: `color(crimson)(sqrt7+sqrt3)``12/(sqrt7sqrt3)` `=` `12/(sqrt7sqrt3) xx color(crimson)((sqrt7+sqrt3)/(sqrt7+sqrt3))` `=` `(12(sqrt7+sqrt3))/((sqrt7sqrt3)(sqrt7+sqrt3))` Simplify using the difference of two squares `=` `(12(sqrt7+sqrt3))/((sqrt7)^2(sqrt3)^2)` Simplify square roots `=` `(12(sqrt7+sqrt3))/(73)` `(sqrt(7))^2 = 7` and `(sqrt(3))^2 = 3` `=` `(12(sqrt7+sqrt3))/4` `=` `(color(crimson)(4:)12(sqrt7+sqrt3))/(4color(crimson)(:4))` Divide top and bottom by `4` `=` `(3(sqrt7+sqrt3))/(class{pkstrike}{4color(crimson)(:4})` Apply the distributive property `=` `3sqrt7+3sqrt3` `sqrt6+sqrt5`
Quizzes
 Simplify Square Roots 1
 Simplify Square Roots 2
 Simplify Square Roots 3
 Simplify Square Roots 4
 Simplify Radicals with Variables 1
 Simplify Radicals with Variables 2
 Simplify Radicals with Variables 3
 Rewriting Entire and Mixed Radicals 1
 Rewriting Entire and Mixed Radicals 2
 Add and Subtract Radical Expressions (Basic) 1
 Add and Subtract Radical Expressions (Basic) 2
 Add and Subtract Radical Expressions (Basic) 3
 Add and Subtract Radical Expressions 1
 Add and Subtract Radical Expressions 2
 Add and Subtract Radical Expressions 3
 Multiply Radical Expressions 1
 Multiply Radical Expressions 2
 Multiply Radical Expressions 3
 Multiply Radical Expressions 4
 Divide Radical Expressions 1
 Divide Radical Expressions 2
 Divide Radical Expressions 3
 Multiply and Divide Radical Expressions
 Simplify Radical Expressions using the Distributive Property 1
 Simplify Radical Expressions using the Distributive Property 2
 Simplify Radical Expressions using the Distributive Property 3
 Simplify Binomial Radical Expressions using the FOIL Method 1
 Simplify Binomial Radical Expressions using the FOIL Method 2
 Rationalizing the Denominator 1
 Rationalizing the Denominator 2
 Rationalizing the Denominator 3
 Rationalizing the Denominator 4
 Rationalizing the Denominator using Conjugates