Lesson 4: Multiply and Divide Integers

New Concept

Mastering integer operations is all about the signs! Whether you multiply or divide, if the signs are the same, your answer is positive. If the signs are different, your answer is negative. These simple rules unlock complex calculations.

What’s next

Next, you'll apply these rules through practice cards, simplify expressions with step-by-step examples, and tackle challenge problems involving variables.

Property

The sign of the product of two numbers depends on their signs. For multiplication of two signed numbers, when the signs are the same, the product is positive, and when the signs are different, the product is negative.

Examples

• To multiply $-8 \cdot 4$, the signs are different, so the product is negative. The result is $-32$.
• To multiply $-6(-9)$, the signs are the same, so the product is positive. The result is $54$.
• To multiply $11 \cdot 5$, the signs are the same, so the product is positive. The result is $55$.

Explanation

If the signs of the two numbers are the same, the product is always positive.
If the signs are different, the product is always negative. Think of it as 'like signs positive, unlike signs negative'.

Property

Multiplying a number by $-1$ gives its opposite.

Examples

• To multiply $-1 \cdot 9$, you get the opposite of $9$, which is $-9$.
• To multiply $-1(-15)$, you get the opposite of $-15$, which is $15$.
• If you have the number $x$, multiplying it by $-1$ gives you $-x$.

Explanation

Multiplying any number by $-1$ is a shortcut to find its opposite.
It flips the number to the other side of zero on the number line, changing its sign from positive to negative or from negative to positive.

Property

Division is the inverse operation of multiplication. The sign of the quotient of two numbers depends on their signs, following the same rules as multiplication.

Examples

• To divide $-45 \div 5$, the signs are different, so the quotient is negative. The result is $-9$.
• To divide $-60 \div (-10)$, the signs are the same, so the quotient is positive. The result is $6$.
• You can check the first example by multiplying back: $-9 \cdot 5 = -45$.

Explanation

The rules for dividing integers are exactly the same as for multiplying!
If the signs match, the answer is positive. If they don't match, the answer is negative. You can always check your answer by multiplying.

Property

Dividing a number by $-1$ gives its opposite.

Examples

• To divide $25 \div (-1)$, you get the opposite of $25$, which is $-25$.
• To divide $-18 \div (-1)$, you get the opposite of $-18$, which is $18$.
• If you have a number $y$, dividing it by $-1$ gives you its opposite, $-y$.

Explanation

Just like with multiplication, dividing a number by $-1$ flips its sign.
A positive number becomes negative, and a negative number becomes positive. It's another quick way to find the opposite of a number.

Property

The placement of parentheses changes the meaning of an expression with a negative base. In $(-a)^n$, the base is $-a$ and is multiplied $n$ times. In $-a^n$, the base is $a$, the exponent is applied first, and then the opposite is taken.

Examples

• To simplify $(-3)^2$, the base is $-3$. You calculate $(-3)(-3) = 9$.
• To simplify $-3^2$, the base is $3$. You calculate $3^2=9$ first, and then take the opposite, giving $-9$.
• For an odd power, the results can be the same: $(-2)^3 = (-2)(-2)(-2) = -8$ and $-2^3 = -(2 \cdot 2 \cdot 2) = -8$.

Explanation

Pay close attention to parentheses! They tell you what gets repeated.
$(-3)^2$ means $(-3)(-3) = 9$. But $-3^2$ means $-(3 \cdot 3) = -9$. The parentheses are powerful!

Property

To evaluate an expression for a given integer value, substitute the integer for the variable. Use parentheses when substituting, especially for negative numbers, to ensure correct calculations according to the order of operations.

Examples

• Evaluate $4x^2 - 2x + 5$ when $x = -3$. Substitute to get $4(-3)^2 - 2(-3) + 5 = 4(9) - (-6) + 5 = 36 + 6 + 5 = 47$.
• Evaluate $5a + 2b - 1$ when $a = -2$ and $b = 4$. Substitute to get $5(-2) + 2(4) - 1 = -10 + 8 - 1 = -3$.
• Evaluate $10 - y^3$ when $y = -2$. Substitute to get $10 - (-2)^3 = 10 - (-8) = 10 + 8 = 18$.

Explanation

When you plug a number into a formula, especially a negative one, wrap it in parentheses.
This protects the negative sign and helps you follow the order of operations (PEMDAS) correctly, avoiding common mistakes with signs.