Example 2.1.13

Evaluate \(x^2\) if \(x=-2\text{.}\)

We substitute:

\begin{align*} x^2\amp=(\substitute{-2})^2\\ \amp=4 \end{align*}

If we don't use parentheses, we would have:

\begin{align*} x^2\amp=-2^2\amp\text{incorrect!}\\ \amp=-4 \end{align*}

The original expression takes \(x\) and squares it. With \(-2^2=-4\text{,}\) the number \(-2\) is not being squared. Since the exponent has higher priority than the negation, it's just the number \(2\) that is being squared. With \((-2)^2=4\) the number \(-2\) is being squared, which is what we would want given the expression \(x^2\text{.}\)

So it is wise to always use some parentheses when substituting in any negative number.