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The fourth root of a number is a value that, when multiplied by itself four times, gives the original number. In mathematical terms, the fourth root of a number \( a \) is expressed as \( a^{1/4} \) or \( \sqrt[4]{a} \). Fourth roots are a special type of root that involve an even index.

For any even root, the expression under the root (known as the radicand) must be non-negative. This is because there are no real roots for negative numbers under an even root index. Therefore, to be a real number, the radicand must be \( \geq 0 \) so the fourth root can be calculated. For example, if we consider the fourth root expression \( \sqrt[4]{x} \), \( x \) must be a non-negative number.

Inequalities are mathematical expressions that involve comparison between two values. They are written using symbols such as \(<\), \(>\), \(\leq\), and \(\geq\). In the case of a function or expression involving a fourth root like \( \sqrt[4]{\frac{1-x}{2+x}} \), we use inequalities to determine when the expression inside the root is non-negative.

This is essential because, as discussed earlier, a fourth root requires a non-negative radicand for the result to be a real number.

By setting an inequality \( \frac{1-x}{2+x} \geq 0 \), we filter out the domain of \( x \) for which the expression is valid. Understanding and solving such inequalities allow us to determine intervals where specific mathematical conditions hold true.

The domain of a function consists of all possible input values (or \( x \)-values) that the function can accept. For a function to be defined, each element in its domain should not lead to mathematical indefinites such as division by zero or taking an even root of a negative number.

The domain is determined by analyzing conditions imposed by the function's formula. In the fourth root expression \( \sqrt[4]{\frac{1-x}{2+x}} \), finding the domain involves ensuring that the radicand \( \frac{1-x}{2+x} \) stays non-negative and does not cause the denominator to equal zero.

By solving the inequality \( \frac{1-x}{2+x} \geq 0 \) and examining when the denominator \( 2+x \) becomes zero, we correctly identify the intervals or values where the function holds true, giving us the domain: \( -2 < x \leq 1 \).

When dealing with roots, and especially even roots like the square root or fourth root, the condition of the radicand (the number inside the root) being non-negative is crucial for the expression to result in a real number.

The radicand cannot be negative as even roots of negative numbers aren't real in the field of real numbers; they are complex. Therefore, when you see an expression like \( \sqrt[4]{a} \), you ensure \( a \geq 0 \).

Applying this to our problem, with the expression \( \sqrt[4]{\frac{1-x}{2+x}} \), means establishing the criterion \( \frac{1-x}{2+x} \geq 0 \). This ensures that the radicand is non-negative across all accepted values of \( x \), reflecting a valid interval in its domain, ensuring our calculations yield real numbers.