Radical Expression Value

This solution explains how to find the value of a given algebraic expression involving square roots.

Expression Evaluation Details

Given the variable $x = \frac{\sqrt{3}}{2}$, find the value of the expression $\sqrt{1 + x} + \sqrt{1 - x}$.

Step-by-Step Calculation

Let the expression be denoted by $E$.
$E = \sqrt{1 + x} + \sqrt{1 - x}$

Squaring the Expression

Squaring both sides simplifies the expression:
$ E^2 = \left( \sqrt{1 + x} + \sqrt{1 - x} \right)^2 $ Using the binomial expansion $(a+b)^2 = a^2 + b^2 + 2ab$:
$ E^2 = (1 + x) + (1 - x) + 2 \sqrt{(1 + x)(1 - x)} $ Simplify the terms:
$ E^2 = 2 + 2 \sqrt{1 - x^2} $

Substituting x Value

Substitute the given value $x = \frac{\sqrt{3}}{2}$ into the equation for $E^2$.
First, calculate $x^2$:
$ x^2 = \left( \frac{\sqrt{3}}{2} \right)^2 = \frac{3}{4} $

Substitute $x^2 = \frac{3}{4}$ back into the $E^2$ equation:
$ E^2 = 2 + 2 \sqrt{1 - \frac{3}{4}} $
$ E^2 = 2 + 2 \sqrt{\frac{4 - 3}{4}} $
$ E^2 = 2 + 2 \sqrt{\frac{1}{4}} $
$ E^2 = 2 + 2 \left( \frac{1}{2} \right) $
$ E^2 = 2 + 1 $
$ E^2 = 3 $

Determining Final Value

Since $E^2 = 3$, taking the square root yields $E = \pm \sqrt{3}$.
The original expression $E = \sqrt{1 + x} + \sqrt{1 - x}$ involves the sum of two positive square roots (because $1 + \frac{\sqrt{3}}{2}$ and $1 - \frac{\sqrt{3}}{2}$ are both positive). Thus, $E$ must be positive.
Therefore, $E = \sqrt{3}$.