How do you simplify #sqrt(1+x) - sqrt(1-x)#?

Question

Abigail Allen · Accepted Answer

#sqrt(1+x) - sqrt(1-x)# does not really simplify,

but you can re-express it in various ways...

First note that for both square roots to have Real values, we must have #x in [-1, 1]# Let's see what happens when you square #sqrt(1+x) - sqrt(1-x)# ... #(sqrt(1+x) - sqrt(1-x))^2# #= (sqrt(1+x))^2 - 2(sqrt(1+x))(sqrt(1-x)) + (sqrt(1-x))^2# #= (1+x) - 2sqrt(1-x^2) + (1-x)# [[ using #sqrt(a)sqrt(b) = sqrt(ab)# ]] #= 2 - 2sqrt(1-x^2)# So #sqrt(1+x) - sqrt(1-x) = +-sqrt(2 - 2sqrt(1-x^2))# What is the correct sign to choose? If #x >= 0# then #1 + x >= 1 - x#, so #sqrt(1+x)-sqrt(1-x) >= 0# If #x < 0# then #1 + x < 1 - x#, so #sqrt(1+x)-sqrt(1-x) < 0# So we have: #sqrt(1+x) - sqrt(1-x) = { (sqrt(2 - 2sqrt(1-x^2)), "if " x >= 0), (-sqrt(2 - 2sqrt(1-x^2)), "if " x < 0) :}# If you like, you can separate out the common factor #2# to get: #sqrt(1+x) - sqrt(1-x) = { (sqrt(2)sqrt(1 - sqrt(1-x^2)), "if " x >= 0), (-sqrt(2)sqrt(1 - sqrt(1-x^2)), "if " x < 0) :}# graph{sqrt(1+x)-sqrt(1-x) [-5, 5, -2.5, 2.5]}

Jace Anderson · Answer

undefined To simplify sqrt(1+x) - sqrt(1-x), we can use the difference of squares formula. By multiplying the expression by its conjugate, we can eliminate the square roots.

The conjugate of sqrt(1+x) - sqrt(1-x) is sqrt(1+x) + sqrt(1-x).

Multiplying the expression by its conjugate, we get:

(sqrt(1+x) - sqrt(1-x)) * (sqrt(1+x) + sqrt(1-x))

Using the difference of squares formula, this simplifies to:

(1+x) - (1-x)

Simplifying further, we have:

1 + x - 1 + x

Combining like terms, the final simplified expression is:

2x