We will learn the sign rules for multiplication（multiplication）and division（division）. We will also learnpowers, which represent multiplying the same number repeatedly, andreciprocals, which are important in division.
Sign Rules for Multiplication and Division
For multiplication and division of two numbers, the sign is determined as follows.
Same signs → positive（＋）、different signs → negative（−）.
- (+)×(+) ＝ ＋ (−)×(−) ＝ ＋
- (+)×(−) ＝ − (−)×(+) ＝ −
Multiply（or divide）the absolute values normally. In a product of three or more numbers, an even number of negative numbers → positive; an odd number → negative.
Calculate the following.
(1) (−6)×(+4) (2) (−15)÷(−3) (3) (−2)×(+3)×(−5)
Answer：
(1) Different signs → negative, 6×4=24 → −24
(2) Same signs → positive, 15÷3=5 → +5
(3) Two negative numbers（even）→ positive, 2×3×5=30 → +30
Powers
A product formed by multiplying the same number repeatedly is called apowerand is written with a small number（theexponent）in the upper right.
2×2×2 ＝ 2³（read “2 to the third power”）, 2 is thebase, and 3 is the exponent.
(−3)² ＝ (−3)×(−3) ＝ +9 ／ (−3)³ ＝ (−3)×(−3)×(−3) ＝ −27
Note：The meaning changes depending on the position of the parentheses.
・(−3)² ＝ (−3)×(−3) ＝ +9 … −3 squared
・−3² ＝ −(3×3) ＝ −9 … 3 squared with a minus sign
Reciprocals and Division
When the product of two numbers is 1, either number is called thereciprocalof the other.
Example：The reciprocal of 2 is 1/2. The reciprocal of −3/4 is −4/3（the sign remains the same）.
Division can be changed intomultiplication by the reciprocal. a÷b ＝ a×(1/b).
Calculate the following.
(1) (−2)³ (2) (−4)÷(2/3)
Answer：
(1) (−2)×(−2)×(−2)：three negative numbers（odd）→ negative, absolute value 2×2×2=8 → −8
(2) The reciprocal of 2/3 is 3/2. (−4)×(3/2) ＝ −12/2 ＝ −6
- Signs in multiplication and division：same signs → positive, different signs → negative
- The number of negative numbers determines the sign of a product：even → positive, odd → negative
- (−a)² and −a² are different! Pay attention to parentheses
- Changing division into multiplication by the reciprocal makes fraction calculations easier
Practice Problems
- Calculate (−8)×(−3)÷(+6).
- Find the values of −5² and (−5)².
- Calculate (−9)÷(−3/4).
Answers and Explanations
- Answer：+4
Explanation：There are two negative numbers（even）, so the sign is positive. Absolute values：8×3÷6=24÷6=4。Therefore, +4。 - Answer：−5² ＝ −25、(−5)² ＝ +25
Explanation：−5² means “change the sign of 5 squared,” so −(5×5)=−25。(−5)² means “square −5,” so (−5)×(−5)=+25。 - Answer：+12
Explanation：The reciprocal of −3/4 is −4/3。 (−9)×(−4/3)。Negative × negative = positive, and the absolute value is 9×4/3 ＝ 36/3 ＝ 12。Therefore, +12。