You already know how to evaluate an expression and how to work with signed numbers. Algebra puts the two together, and that is where small sign slips creep in. One habit prevents most of them: put every substituted value in parentheses.
Why it works
An exponent applies to the thing directly in front of it. In $x^2$ that thing is $x$, so when $x = -3$ the whole value $-3$ is squared: $x^2 = (-3)(-3) = 9$. But in $-3^2$ the exponent sits on the $3$ alone, and the minus sign means “the opposite of”: $-3^2 = -(3 \cdot 3) = -9$. Parentheses keep the value in one piece, so the exponent acts on all of it.
| Written | Means | Value |
|---|---|---|
| $(-3)^2$ | $(-3)(-3)$ | $9$ |
| $-3^2$ | $-(3 \cdot 3)$ | $-9$ |
| $-(-3)^2$ | $-\big((-3)(-3)\big)$ | $-9$ |
| $(-2)^3$ | $(-2)(-2)(-2)$ | $-8$ |
Key idea
Substitute each value in parentheses, then follow the order of operations: grouping (parentheses and fraction bars), exponents, multiplication and division from left to right, then addition and subtraction from left to right.
Worked examples
Worked example: A quadratic at a negative number
Evaluate $x^2 - 4x + 1$ when $x = -3$.
- Substitute in parentheses: $(-3)^2 - 4(-3) + 1$.
- Exponent first: $(-3)^2 = 9$.
- Multiply: $-4(-3) = 12$.
- Add from left to right: $9 + 12 + 1 = 22$.
Answer: $22$
Worked example: A minus sign in front of a power
Evaluate $-x^2 + 2xy$ when $x = -2$ and $y = 5$.
- Substitute: $-(-2)^2 + 2(-2)(5)$.
- Exponent first: $(-2)^2 = 4$, so the first term is $-4$.
- Multiply: $2(-2)(5) = -20$.
- Add: $-4 + (-20) = -24$.
Answer: $-24$
Worked example: A fraction bar groups
Evaluate $\dfrac{a^2 - b}{a + b}$ when $a = -4$ and $b = 2$.
- The fraction bar works like parentheses around the top and around the bottom.
- Top: $(-4)^2 - 2 = 16 - 2 = 14$.
- Bottom: $-4 + 2 = -2$.
- Divide: $\frac{14}{-2} = -7$.
Answer: $-7$
Common mistake
Substituting $x = -3$ into $x^2$ without parentheses gives $-3^2 = -9$, which is wrong: a square of any real number is never negative. Check: if you squared a value and got a negative answer, a parenthesis is missing.