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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)(a-b)=a^2-b^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)(2-sqrt2)``1/(2+sqrt2)` `=` `1/(2+sqrt2) xx color(crimson)((2-sqrt2)/(2-sqrt2))` `=` `(2-sqrt2)/((2+sqrt2)(2-sqrt2))` Simplify using the difference of two squares `=` `(2-sqrt2)/((2)^2-(sqrt2)^2)` Simplify `=` `(2-sqrt2)/(4-2)` `(2)^2 = 4` and `(sqrt(2))^2 = 2` `=` `(2-sqrt2)/2` `(2-sqrt2)/2` -
Question 2 of 4
2. Question
Express the following with rational denominators:
`1/(sqrt6-sqrt5)`
Correct
Excellent!
Incorrect
Difference of Two Squares
`(a+b)(a-b)=a^2-b^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/(sqrt6-sqrt5)` `=` `1/(sqrt6-sqrt5) xx color(crimson)((sqrt6+sqrt5)/(sqrt6+sqrt5))` `=` `(sqrt6+sqrt5)/((sqrt6-sqrt5)(sqrt6+sqrt5))` Simplify using the difference of two squares `=` `(sqrt6+sqrt5)/((sqrt6)^2-(sqrt5)^2)` Simplify square roots `=` `(sqrt6+sqrt5)/(6-5)` `(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/(4-sqrt2)`
Correct
Excellent!
Incorrect
Difference of Two Squares
`(a+b)(a-b)=a^2-b^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/(4-sqrt2)` `=` `4/(4-sqrt2) xx color(crimson)((4+sqrt2)/(4+sqrt2))` `=` `(4 xx 4+4sqrt2)/((4-sqrt2)(4+sqrt2))` Simplify using the difference of two squares `=` `(16+4sqrt2)/(4^2-(sqrt2)^2)` Simplify square roots `=` `(16+4sqrt2)/16-2` `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/(sqrt7-sqrt3)`
Correct
Excellent!
Incorrect
Difference of Two Squares
`(a+b)(a-b)=a^2-b^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/(sqrt7-sqrt3)` `=` `12/(sqrt7-sqrt3) xx color(crimson)((sqrt7+sqrt3)/(sqrt7+sqrt3))` `=` `(12(sqrt7+sqrt3))/((sqrt7-sqrt3)(sqrt7+sqrt3))` Simplify using the difference of two squares `=` `(12(sqrt7+sqrt3))/((sqrt7)^2-(sqrt3)^2)` Simplify square roots `=` `(12(sqrt7+sqrt3))/(7-3)` `(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{pk-strike}{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