How to simplify sqrt(2 + sqrt 3) + sqrt(2 - sqrt 3)?
- print Print
- list Cite
Expert Answers
calendarEducator since 2010
write12,550 answers
starTop subjects are Math, Science, and Business
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
Related Questions
- lim x->infinity ((sqrt(x^2+5))-(sqrt(x^2+3)))
- 1 Educator Answer
- Simplify (x-2)^2
- 1 Educator Answer
- The integral of dx/x^2 sqrt(x^2 + 9) from sqrt(3) to cube root of 3Using trigonometric substitution
- 1 Educator Answer
- `int_(sqrt(2)/3)^(2/3) (dx)/(x^5 sqrt(9x^2 - 1))` Evaluate the integral
- 1 Educator Answer
- Simplify`sqrt(18)/(sqrt(8)-3)` , `(4-3sqrt(2))^2` and `*sqrt(5b ^2d)`
- 1 Educator Answer
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).
Student Answers