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How to simplify sqrt(2 + sqrt 3) + sqrt(2 - sqrt 3)?

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sandeepjangir | Student, Grade 11 | eNotes Newbie

Posted July 15, 2011 at 12:06 AM via web

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How to simplify sqrt(2 + sqrt 3) + sqrt(2 - sqrt 3)?

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giorgiana1976 | College Teacher | Valedictorian

Posted July 15, 2011 at 12:24 AM (Answer #1)

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To calculate this expression, we'll have to use the identities:

sqrt[a+(sqrtb)] = sqrt{[a+sqrt(a^2 - b)]/2} + sqrt{[a-sqrt(a^2 - b)]/2}

sqrt[a-(sqrtb)] = sqrt{[a+sqrt(a^2 - b)]/2} - sqrt{[a-sqrt(a^2 - b)]/2}

Let a = 2 and b = 3

sqrt[2+(sqrt3)] = sqrt{[2+sqrt(2^2 - 3)]/2} + sqrt{[2-sqrt(2^2 - 3)]/2}

sqrt[2+(sqrt3)] = sqrt(3/2) + sqrt(1/2) (1)

sqrt[2-(sqrt3)] = sqrt{[2+sqrt(2^2 - 3)]/2} - sqrt{[2-sqrt(2^2 - 3)]/2}

sqrt[2+(sqrt3)] = sqrt(3/2) - sqrt(1/2) (2)

We'll add (1) and (2) and we'll get:

sqrt[2+(sqrt3)] + sqrt[2-(sqrt3)] = sqrt(3/2) + sqrt(1/2) + sqrt(3/2) - sqrt(1/2)

We'll eliminate like terms:

sqrt[2+(sqrt3)] + sqrt[2-(sqrt3)] = 2*sqrt(3/2)

The requested result of the expression is sqrt[2+(sqrt3)] + sqrt[2-(sqrt3)] = 2*sqrt(3/2).

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justaguide | College Teacher | (Level 1) Adjunct Educator

Posted July 15, 2011 at 1:08 AM (Answer #2)

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We have to simplify sqrt(2 + sqrt 3) + sqrt(2 - sqrt 3)

Let A = sqrt(2 + sqrt 3) + sqrt(2 - sqrt 3)

A^2 = (sqrt(2 + sqrt 3) + sqrt(2 - sqrt 3))^2

=> 2 + sqrt 3 + 2 - sqrt 3 + 2*sqrt(2 + sqrt 3)sqrt(2 - sqrt 3)

=> 4 + 2*(2^2 - (sqrt 3)^2)

=> 4 + 2*(4 - 3)

=> 6

A = sqrt 6

sqrt(2 + sqrt 3) + sqrt(2 - sqrt 3) = sqrt 6

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