These questions ask you to find products, quotients and missing factors while keeping the sign separate from the size. Some also combine several negative factors or write a negative number squared or cubed.
Decide the sign, then calculate the size
Use two passes. First ignore the sizes and decide whether the result is above zero, below zero or exactly zero. Then multiply or divide the positive magnitudes. For two nonzero numbers, matching signs give a positive result and different signs give a negative result. For example, (-6) times 3 has different signs, so its sign is negative; 6 times 3 has magnitude 18, giving -18. By contrast, (-20) divided by (-5) has matching signs, so the quotient is positive 4.
| Signal | Decision |
|---|---|
| Two nonzero factors or a division | Matching signs mean positive; different signs mean negative. |
| Several nonzero factors | Count negative factors. An even count means positive; an odd count means negative. |
| A zero factor | The product is zero before any other sign decision is needed. |
| A missing factor | Use the required product sign to choose the missing sign, then divide magnitudes. |
When multiplying and dividing appear in one line, they have equal rank, so work from left to right while carrying the sign at every stage. For powers written as repeated multiplication, count the copies of the negative factor. Negative three squared means two copies of -3 and is positive 9; negative three cubed means three copies and is -27. The words and parentheses make the sign part of every factor.
Sign traps to avoid
- Carrying a minus sign through two negative numbers stops after only one sign reversal. Check whether the two signs match before doing the magnitude arithmetic.
- Looking only at the final factor in a long product ignores earlier sign changes. Count all negative factors and pair them; only an unpaired negative leaves the product below zero.
- Finishing the multiplication but forgetting a following division leaves an intermediate value instead of the value of the whole expression. Move through equal-rank operations from left to right.
- Doubling a base is not the same as squaring it. Squared means multiply two copies, while cubed means multiply three copies, with the negative sign included in every parenthesised copy.
- Using addition or subtraction on the two visible magnitudes does not answer a multiplication or division. Decide the sign, then perform the operation actually shown.
Try it
Work out 6 * (-4). Enter the product.
In each of 3 rounds of a game, 2 players each lose 5 points. The total change in points is 2 * (-5) * 3. What is the total change?
- -10
- 30
- -15
- -30
One factor is -7 and the product is 56. Enter the other factor.