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Multiplying Monomials

Monomials

A monomial is a number, a variable, or a product of a number and one or more variables.

The only rules are that the variables should be raised to only positive integer powers (no square roots or $\frac{1}{x}$ 's allowed), and no plus or minus signs.

Examples:

$17$ , $3 x y$ , $- 4 x^{2}$ ,

$\frac{7}{100} a^{5} b^{6} c^{7} d^{8} z^{999}$

Multiplication of Monomials

When you multiply monomials, first multiply the coefficients and then multiply the variables by adding the exponents.

Examples:

$\begin{array}{l} 1) 4^{2} \cdot 4^{3} = 4^{2 + 3} \\ = 4^{5} \\ 2) x^{2} \cdot x^{5} = x^{2 + 5} \\ = x^{7} \\ 3) 4^{2} \cdot x^{5} = 16 \cdot x^{5} \\ = 16 x^{5} \\ 4) (- 2 a^{2} b) (5 a^{3} b^{2}) = (- 2 \cdot 5) (a^{2 + 3}) (b^{1 + 2}) \\ = - 10 a^{5} b^{3} \\ 5) (- 2 a^{2} b) (5 a^{3} b^{2}) (a b c) = (- 2 \cdot 5) (a^{2 + 3 + 1}) (b^{1 + 2 + 1}) (c) \\ = - 10 a^{6} b^{4} c \end{array}$

Note that when you multiply monomials with same base, you can add their exponents.

$a^{m} \cdot a^{n} = a^{m + n}$

This is called the Product of Powers Property.