Overview
An expression records a sequence of operations. Grouping and exponents must be resolved before multiplication; addition and subtraction then proceed left to right. A negative base needs parentheses if the exponent is meant to apply to its sign.
Consider this example: Evaluate −4 + 3(2 − 7). The parentheses give −5. Multiplying by 3 gives −15, so −4 + (−15) = −19. The choice −11 comes from adding 4 to −15, which reverses the sign of the initial −4.
The method also handles a different case: Evaluate −3² + (−3)² + 8 ÷ (−2). The first power is −9; the parenthesized power is 9. They cancel. The remaining quotient is −4. Treating both powers as 9 would give 14.
For a further application, consider: Which value of k makes −2[k − (−4)] = 10 true? Divide both sides by −2 to obtain k + 4 = −5. Thus k = −9. Substitution gives −2(−5) = 10; k = −1 instead gives −6.
Treating a minus sign outside a power as part of the base changes the expression. Writing each intermediate value makes that mistake visible. On a new problem, mark the grouping first and predict the sign before calculating. Check a subtraction by rewriting it as addition of the opposite.
The order rule also matters when division and multiplication occur together: they share precedence, so work from left to right. In 18 ÷ 3 × (−2) − (−5), divide 18 by 3 to get 6, multiply by −2 to get −12, and subtracting −5 adds 5, giving −7. Grouping 3 × (−2) into a denominator would introduce parentheses that were never given. To check a contextual change, keep the starting value separate from the signed changes. A freezer starts at −6°C, rises 9°C, then falls twice that rise. The rise reaches 3°C; twice 9°C is an 18°C fall, so the final temperature is −15°C. Returning to −6°C would undo only one rise. The units stay degrees Celsius at every step.