Training Signed Numbers Multiplying and Dividing Signed Numbers
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Multiplying and Dividing Signed Numbers

22 min Signed Numbers

Multiplying and Dividing Signed Numbers

Multiplication and division of signed numbers follow a beautifully simple pattern: same signs give a positive result, and different signs give a negative result. Once you memorize this two-word rule, you can handle any product or quotient, no matter how many factors are involved.

When an expression contains more than two factors, the shortcut is to count the negative signs. An even number of negatives produces a positive product; an odd number produces a negative product. This same idea extends to exponents — a negative base raised to an even power is positive, and raised to an odd power is negative.

One classic pitfall deserves special attention: the difference between $-3^2$ and $(-3)^2$. Without parentheses, the exponent applies only to the 3, giving $-9$; with parentheses, the entire $-3$ is squared, giving $9$. Mastering these details now will save you from sign errors in every future math course.

Sign Rules
SignsResult
$(+)(+)$ or $(-)(-)$Positive
$(+)(-)$ or $(-)(+)$Negative

Same signs → positive. Different signs → negative.

Example 1

$(-6) \times (-4)$

  1. Same signs → positive.
  2. $(-6)(-4) = 24$.
Example 2

$(-3) \times 5$

  1. Different signs → negative.
  2. $(-3)(5) = -15$.
Example 3

$(-20) \div (-5)$

  1. Same signs → positive.
  2. $(-20) \div (-5) = 4$.
Example 4

$18 \div (-3)$

  1. Different signs → negative.
  2. $18 \div (-3) = -6$.

Multiple Factors

Rule

Count the negative signs: even → positive; odd → negative.

Example 5

$(-2)(-3)(-4)$

  1. 3 negatives (odd) → negative.
  2. $2 \times 3 \times 4 = 24$.
  3. Answer: $-24$.

Exponents with Signed Numbers

Critical Distinction

$-3^2 = -(3^2) = -9$ ← exponent applies only to 3

$(-3)^2 = (-3)(-3) = 9$ ← exponent applies to $-3$

Example 6

Evaluate: $(-2)^3$ and $(-2)^4$

  1. $(-2)^3 = -8$ (odd exponent → negative).
  2. $(-2)^4 = 16$ (even exponent → positive).
Interactive Explorer: Multiply & Divide Signed Numbers
A × B = 24
A ÷ B = 1.5
Sign rule: Same signs → Positive
(-A)² vs -(A²): (-(-6))² = 36 vs -((-6)²) = -36

Practice Problems

1. $(-7)(8)$
2. $(-9)(-6)$
3. $36 \div (-4)$
4. $(-48) \div (-6)$
5. $(-2)(3)(-5)$
6. $(-1)^{10}$
7. $-5^2$
8. $(-5)^2$
9. $(-4)(-2)(-1)(-3)$
10. $\dfrac{(-3)(12)}{-9}$
11. $(-3)^3$
12. $(-1)^{55}$
13. $(-10)(0)(-5)$
14. $\dfrac{-72}{-8}$
15. $(-2)^5$
Show Answer Key

1. $-56$

2. $54$

3. $-9$

4. $8$

5. $30$

6. $1$

7. $-25$

8. $25$

9. $24$ (4 negatives → positive)

10. $4$

11. $-27$

12. $-1$

13. $0$

14. $9$

15. $-32$

Adding and Subtracting Signed Numbers Order of Operations with Signed Numbers