Practice Maths

L46 — Multiplying and Dividing Integers

Key Ideas

Key Terms

product
The result of multiplying two or more numbers together.
quotient
The result of dividing one number by another.
sign rule
The rule that determines whether a product or quotient is positive or negative: same signs give a positive result; different signs give a negative result.
power / exponent
Shorthand for repeated multiplication. In (−3)2, the exponent 2 means (−3) is multiplied by itself twice.

Sign rules for × and ÷:

  • Same signs → Positive: (+) × (+) = (+)    (−) × (−) = (+)
  • Different signs → Negative: (+) × (−) = (−)    (−) × (+) = (−)

Powers of negatives:

  • Even power → positive: (−3)2 = 9
  • Odd power → negative: (−2)3 = −8
Hot Tip: Two negatives multiply to give a positive. Think of it as “the enemy of my enemy is my friend” — (−6) × (−5) = +30.

Worked Examples

(−6) × (−5) = 30  (same signs → positive)

4 × (−7) = −28  (different signs → negative)

(−15) ÷ 3 = −5  (different signs → negative)

(−20) ÷ (−4) = 5  (same signs → positive)

(−2)3 = (−2) × (−2) × (−2) = 4 × (−2) = −8

(−3)2 = (−3) × (−3) = 9

Consolidating the Sign Rules

You've seen integer multiplication before (L34). This lesson gives you more practice and extends the rules to powers. The sign rules are the heart of this topic — get these absolutely automatic:

  • Same signs → Positive: (−6) × (−5) = +30. (+4) × (+7) = +28.
  • Different signs → Negative: (+4) × (−7) = −28. (−3) × (+9) = −27.

The same rules work for division — nothing changes. (−15) ÷ (−3) = +5. (−15) ÷ 3 = −5.

Multiple Negatives: Count the Signs

When multiplying or dividing a chain of integers, count how many negative signs there are:

  • Even number of negatives → overall answer is POSITIVE.
  • Odd number of negatives → overall answer is NEGATIVE.

Example: (−2) × (−3) × (−4). Three negatives = odd → negative result. 2 × 3 × 4 = 24, so the answer is −24.

Example: (−2) × (−3) × (−4) × (−1). Four negatives = even → positive. Answer: +24.

Powers: Even vs Odd Exponent

The “count the negatives” rule explains the pattern for powers:

  • (−3)2 = (−3) × (−3) = +9. Two negatives (even) → positive.
  • (−3)3 = (−3) × (−3) × (−3) = (+9) × (−3) = −27. Three negatives (odd) → negative.
  • (−3)4 = +81. Four negatives → positive. And so on.

Pattern: even power of a negative → positive. Odd power of a negative → negative. This is a reliable shortcut.

Remember: The “enemy of my enemy” saying from the Key Ideas tab sums it up well. Two “bads” together make a “good.” (−) × (−) = (+). One bad and one good make a bad. (−) × (+) = (−). Count the negatives to determine the sign of any product or quotient.

Real-Life Application: Temperature Drop Rate

A temperature that drops 3°C per hour for 5 hours: change = (−3) × 5 = −15°C. If the starting temperature was 10°C, the final temperature is 10 + (−15) = −5°C. Integer multiplication precisely models real-world changes.

Common Mistake: Confusing (−3)2 with −32. They are different! (−3)2 = (−3) × (−3) = +9, because the negative is INSIDE the brackets and gets squared. But −32 means −(32) = −9, because the exponent applies to 3 only, and the negative is applied after. Be careful with bracket placement!

  1. Multiply integers

    1. 3 × (−4)
    2. (−5) × 6
    3. (−3) × (−7)
    4. 8 × (−9)
    5. (−4) × (−4)
    6. (−6) × 5
    7. 7 × (−8)
    8. (−10) × (−3)

  2. Divide integers

    1. (−12) ÷ 3
    2. 20 ÷ (−4)
    3. (−18) ÷ (−6)
    4. (−36) ÷ 9
    5. 45 ÷ (−9)
    6. (−24) ÷ (−8)
    7. (−35) ÷ 7
    8. 56 ÷ (−8)

  3. Powers of negative numbers

    1. (−2)2
    2. (−2)3
    3. (−3)2
    4. (−1)5
    5. (−4)2
    6. (−5)3
    7. (−10)2
    8. (−2)4

  4. Predict the sign and check reasoning

    1. Without calculating, state whether (−8) × (−7) is positive or negative. Explain.
    2. Without calculating, state whether (−6) × 5 × (−2) is positive or negative. Explain.
    3. True or false: (−3)4 is negative. Justify.
    4. A student writes (−4) × (−3) = −12. Is this correct? Fix the error.
    5. How many negative factors must you multiply together to get a positive result? Give a rule.

  5. Real-world contexts

    1. The temperature drops 3°C every hour. What is the total change after 5 hours? Write this as multiplication.
    2. A submarine descends at 4 m per minute. After 6 minutes, what is its depth relative to the surface?
    3. A company loses $250 per day. How much does it lose in 8 days? Express as a signed number.
    4. If a bank account loses $60 each week, how many weeks until the balance drops from $0 to −$300?
    5. The temperature was −12°C. It rose by equal amounts over 4 hours to reach 0°C. How many degrees did it rise each hour?

  6. Mixed multiplication and division

    1. (−3) × (−4) × 2
    2. 5 × (−2) × (−3)
    3. (−1) × (−1) × (−1)
    4. (−6) × 0
    5. (−72) ÷ (−8)
    6. (−90) ÷ 9
    7. 108 ÷ (−12)
    8. (−144) ÷ (−12)

  7. Find the missing integer

    1. __ × (−5) = 35
    2. (−4) × __ = −28
    3. __ ÷ (−3) = 9
    4. (−36) ÷ __ = −6
    5. __ × (−8) = 64
    6. (−10) × __ = 0

  8. Order of operations with integers

    1. (−3) × 4 + 2
    2. 6 ÷ (−2) + (−5)
    3. (−8) × (−2) − 10
    4. (−4)2 − 20
    5. 3 × (−3) + (−3) × (−3)
    6. (−2)3 + (−2)2

  9. Worded integer problems

    1. A lift in a building moves 3 floors down every 10 seconds. After 50 seconds, how many floors has it moved? Write this as multiplication of integers.
    2. A scientist records a temperature drop of 4°C per hour. Starting at 0°C, what is the temperature after 6 hours? Is the answer the same if the rate were +4°C per hour for −6 hours? Explain.
    3. A quarterback loses 7 yards on each of 4 consecutive plays. Express the total yardage change as multiplication of integers and give the answer.
    4. A business records profit/loss data: Week 1: −$800, Week 2: −$800, Week 3: −$800, Week 4: −$800. Write the total as repeated multiplication. What is the total for 4 weeks?

  10. Error spotting and reasoning

    1. Sam says: “(−2)3 = 8 because 23 = 8.” What mistake has Sam made? What is the correct answer?
    2. Jess says: “A negative number divided by a negative number is always negative.” Is Jess correct? Give a counterexample if she is wrong.
    3. Three friends each owe $15. Write a multiplication equation to show how much they owe in total as a negative amount. What is the total?
    4. Explain in your own words why multiplying two negative integers gives a positive result. Use a pattern to support your explanation (e.g. (−3) × 3, (−3) × 2, (−3) × 1, …).
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