$-3^2 = 9?\ $ Correct syntax for a negative number with an exponent.

$-x^2$, in every mathematical context I have seen, always means $-(x^2)$. So $-3^2 = -9$.

On the other hand, when you plug in a value to an expression you don't just plug the symbols in directly, you add parentheses first. For example, if you plug in $x = y + 3$ to the expression $7x$, you get $7(y + 3) = 7y + 21$, not $7y + 3$. Similarly, plugging in $x = -3$ to the expression $x^2$ gives $(-3)^2 = 9$, not $-3^2 = -9$. So the book does not contradict itself.

$-3^2$ is always $-9$. There's no ambiguity. And the odd/even rule is also true! $x^n$ is always nonnegative when $n$ is even, and $x^n$ is the same sign as $x$ when $n$ is odd (when $x$ is real).

There's no contradiction, because "$-3^2$" isn't actually of the form $x^n$. See, $x^n$, when you substitute $x=-3$ and $n=2$, gives you "$(-3)^2$," not "$-3^2$." Remember, when you substitute in, you always need parentheses.

Why do we need parentheses? I'll give an example.

• What is $7-x$, when $x=2$?

The answer is $7-2=5$.

And now, the same question, reworded:

• What is $7-x$, when $x=4-2$?

Why is this the same question? Because $4-2$ is the same thing as $2$. That means that we should get the same answer. And yet: $$7-4-2=3-2=1,$$ a different answer! The only way to get the answers to agree is to write $7-(4-2)=7-2=5$. So we see that, in this example, we needed parentheses.

It turns out that when we put in the parentheses, we always get the right answer, and we've just seen that leaving them out can get you the wrong answer. This means that you need to put in the parentheses.

6005 already explained why they don't contradict themselves. As an alternative answer to the first part of the question, "higher order" operators usually take precedence: exponentiation is applied before multiplication, which again is applied before subtraction. Whether you interpret unary minus in $-x$ to be $0-x$ or $(0-1)∙x$ it then follows that $-x^2$ should be calculated as $-(x^2)$.

It's not a matter of syntax, it's a matter of operator precedence. In the absence of parentheses, exponentiation is executed first, then negation. Try this in Wolfram Alpha: -3^2. Then try (-3)^2. We have long agreed on these rules so that computers deliver consistent results on calculations involving various different arithmetic operations.

As for the odd-negative/even-positive thing, that only applies if the base is negative. Observe for example:

• $(-2)^2 = (-2) \times (-2) = 4$
• $(-2)^3 = (-2) \times (-2) \times (-2) = -8$
• $(-2)^4 = (-2) \times (-2) \times (-2) \times (-2) = 16$
• $(-2)^5 = (-2) \times (-2) \times (-2) \times (-2) \times (-2) = -32$

See also this Sloane's A122803.