Lesson 3: Multiplying and Dividing Signed Numbers

New Concept

This lesson introduces the rules for multiplying and dividing signed numbers. You'll learn how the signs of the numbers determine the sign of the product or quotient and apply these rules to evaluate expressions, including powers of negative numbers.

What’s next

Now that you have the basic rules, you'll apply them in worked examples, interactive practice problems, and real-world application challenges.

Property

1. The product of two numbers with the same sign is positive.
2. The product of two numbers with opposite signs is negative.
3. The product of any number and zero is zero.

Examples

• The product of two negative numbers is positive: $(-6)(-7) = 42$.
• The product of a positive and a negative number is negative: $9(-3) = -27$.
• Any number multiplied by zero is zero: $(-15)(0) = 0$.

Explanation

Think of multiplication as repeated addition. Multiplying by a negative number flips the sign. A positive times a negative is negative. A negative times a negative flips the sign back to positive, making the result positive.

Property

1. The quotient of two numbers with the same sign is positive.
2. The quotient of two numbers with opposite signs is negative.
3. Zero divided by any number (except zero) is zero.
4. The quotient of any number divided by zero is undefined.

Examples

• The quotient of two numbers with opposite signs is negative: $32 \div (-8) = -4$.
• The quotient of two numbers with the same sign is positive: $\frac{-45}{-9} = 5$.
• Division by zero is undefined, but zero divided by a non-zero number is zero: $\frac{-14}{0}$ is undefined, while $\frac{0}{7} = 0$.

Explanation

Division is the inverse of multiplication, so its sign rules are the same. Since $3 \times (-4) = -12$, it follows that $-12 \div (-4) = 3$. Division by zero is undefined because no number can multiply by zero to get a non-zero result.

Property

To show that a negative number is raised to a power, we must enclose the negative number in parentheses.

Examples

• When the negative base is in parentheses, an even exponent results in a positive number: $(-3)^4 = (-3)(-3)(-3)(-3) = 81$.
• Without parentheses, the exponent applies only to the number, and the negative sign is applied last: $-3^4 = -(3 \cdot 3 \cdot 3 \cdot 3) = -81$.
• An odd exponent on a negative base always results in a negative number: $(-2)^3 = (-2)(-2)(-2) = -8$.

Explanation

Parentheses are crucial! They tell you what the base of the exponent is. With parentheses, like in $(-3)^2$, the base is $-3$. Without them, as in $-3^2$, the base is just $3$, and the negative sign is applied after.