Lesson 36: Multiplying and Dividing Integers and Multiplying and Dividing Terms

New Concept

The new rule for multiplying and dividing integers concerns their signs: If the two numbers have the same sign, the answer is positive. If the two numbers have different signs, the answer is negative.

What’s next

Now, let's apply this. We'll solve problems involving exponents and variables, and see how the rule works when expanding algebraic expressions.

Property

If the two numbers have the same sign, the answer is positive. If the two numbers have different signs, the answer is negative.

Examples

$(-8) \times (-4) = 32$
$\frac{-20}{5} = -4$
$(6)(-7) = -42$

Explanation

Think of it like this: a friend (+) of a friend (+) is a friend (+). An enemy (-) of an enemy (-) is also a friend (+). But a friend (+) of an enemy (-) is an enemy (-). Same signs are positive pals; different signs are negative news! This helps you remember the rule for any multiplication or division problem.

Property

Squaring a negative number results in a positive number. Cubing a negative number results in a negative number. An even number of negative factors is positive, and an odd number of negative factors is negative.

Examples

$(-3)^2 = (-3)(-3) = 9$
$(-3)^3 = (-3)(-3)(-3) = -27$
$(-1)^{10} = 1$

Explanation

When you raise a negative number to a power, it’s all about pairing up the negative signs. If the exponent is an even number, every negative sign finds a partner and they turn positive together. But if the exponent is odd, one lonely negative sign is left over, making the final answer negative every single time.

Property

Multiplication: Use the commutative and associative properties to rearrange and regroup the factors. Division: Factor the dividend and divisor and then reduce.

Examples

$(7x^3y)(-3xyz) = (7)(-3)(x^3 \cdot x)(y \cdot y)(z) = -21x^4y^2z$
$\frac{15a^4b^2}{-3ab} = \frac{3 \cdot 5 \cdot a \cdot a \cdot a \cdot a \cdot b \cdot b}{-1 \cdot 3 \cdot a \cdot b} = -5a^3b$
$\frac{-24x^2y^3z}{-8x^2y} = (\frac{-24}{-8}) (\frac{x^2}{x^2}) (\frac{y^3}{y}) (z) = 3y^2z$

Explanation

This is like sorting your laundry. First, handle the numbers (coefficients) by multiplying or dividing them. Then, group the same variables together. For multiplication, combine them by adding their exponents. For division, cancel out common factors from the top and bottom. This process makes complex algebraic expressions neat and tidy, just like perfectly folded clothes.