lesson
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Imagine designing a video game screen where a character's speed is 2x and the terrain friction scales by 3x2. What happens when those two quantities multiply together?
To find out, we multiply monomials — single algebraic terms formed by multiplying numbers and variables together, like 4x3 or −7ab2.
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Before combining numbers and variables together, what happens when we multiply just the variable parts, like x2⋅x3?
The Product Rule for Exponents
An exponent counts repeated multiplication: x2 means x⋅x, and x3 means x⋅x⋅x.
When you multiply them, you get (x⋅x)⋅(x⋅x⋅x), which is 5 factors of x multiplied together: x5.
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In 1637, French philosopher and mathematician René Descartes popularized this raised exponent notation in his work La Géométrie to avoid writing repetitive strings of variables.
This pattern holds for any powers with matching bases, giving us the Product Rule: xa⋅xb=xa+b.
Now, what happens when our terms also have numbers attached out front?
Multiplying Coefficients and Variables
To multiply monomials, use the Commutative and Associative Properties of Multiplication to rearrange and group the numerical coefficients together and like variable bases together: (a⋅xm)(b⋅xn)=(a⋅b)(xm⋅xn).