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

  1. \(\sqrt{y} - 2\)

    Because the index is \(2\) (even), \(y \ge 0, y \in \rm R\).

  2. \(\sqrt{2w - 3} + 10\)

    Because the index is \(2\) (even), \(2w - 3 \ge 0\).

    \(\begin{align}
     2w - 3 &\ge 0 \\
     2w &\ge 3 \\
     w &\ge \frac{3}{2},w \in \rm R \\
     \end{align}\)

  3. \(\sqrt[5]{2x + 6}\)

    Because the index is \(5\) (odd), there are no restrictions on the variable, \(x \in \rm R\).

  4. \(\sqrt{d^2 + 1} - 3\)

    Because the index is \(2\) (even), the radicand must be greater than or equal to zero. Because \(d\) is squared, \(d^2 \ge 0\) , adding \(1\) will make the radicand always greater than zero. Therefore, there are no restrictions on the variable, \(d \in \rm R\).

    For further information about restrictions on variables in radicals, see p. 273 of Pre-Calculus 11.