Example  1
State the restrictions on the variable for the following radical expressions.

  1. \(\sqrt{x}\)

    Because the index is \(2\) (even), the radicand must be greater than or equal to zero. Or, \(x \ge 0, x \in \rm \thinspace R\).

  2. \(\sqrt[3]{x} - 24\)

    Because the index is \(3\) (odd), there are no restrictions on the radicand.  \(x \in \rm \thinspace R\)
  3. \(\sqrt{x - 1} + \sqrt{x}\)

    Because the index is \(2\) (even), the radicands must be greater than or equal to zero. Note that for this question there are two radicals to consider separately.

    Step 1: Determine the restrictions on each radical.

    \(\begin{align}
     x - 1 &\ge 0 \\
     x &\ge 1 \\
     \end{align}\)
    \(x \ge 0\)

    Step 2: Simplify so that both restrictions are included in one.

    This is easiest to see using a number line. Look for the areas of overlap.



    This indicates that the restriction on the variable is \(x \ge 1, x \in \rm \thinspace R\). Note that restricting the variable to only \(x\) greater than or equal to zero would be insufficient for the entire expression.
  4. \(\sqrt[5]{x + 3}\)

    Because the index is \(5\) (odd), there are no restrictions on the radicand. \(x \in \rm \thinspace R\)
  5. \(\frac{7}{\sqrt{x}}\)

    Because the index is \(2\) (even), the radicand must be greater than or equal to zero. However, because \(x\) is in the denominator, it cannot equal zero either. \(x > 0, x \in \rm \thinspace R\)