The following are rules regarding the multiplying of variable expressions.

Rule 1: To multiply monomials with the same base, keep the base and add the powers:

x ^{a}x ^{b}= x ^{a}^{ + }^{b}

Rule 2: To raise a base to a power, keep the base and multiply the powers.
 ( x ^{a}) ^{b}= x ^{ab}

Rule 3: To raise a product to a power, raise each factor in the product to that power.
 ( xy) ^{a}= x ^{a}y ^{a}
Example 1
Simplify each of the following multiplication problems and state which of the preceding rules was applied.

yy ^{5}
 ( x ^{4}) ^{3}
 (–2 x ^{4} y ^{2} z ^{3}) ^{5}

a ^{3}( a ^{2} b ^{3}) ^{4}




To multiply monomials together, follow this procedure.
 Multiply the numerical coefficients together.
 Multiply the variables together.
 Write the results as a product.
Example 2
Simplify each of the following.
 (4 x ^{2})(3 x ^{3})
 (–8 a ^{3} b ^{2})(2 a ^{2} b ^{2}) ^{3}
 (4 x ^{2})(3 x ^{3}) = (4 × 3)( x ^{2} x ^{3}) = 12 x ^{5}
 (–8 a ^{3} b ^{2})(2 a ^{2} b ^{2}) ^{3} = (–8 a ^{3} b ^{2})(8 a ^{6} b ^{6}) = –64 a ^{9} b ^{8}
To multiply polynomials together, multiply each term in one polynomial by each term in the other polynomial. Then simplify if possible.
Example 3
Multiply each of the following.
 5 x(3 x ^{2} – 4 x + 2)
 (4 x – 2)(3 x + 5)
 ( x + y)( x ^{2} – xy + y ^{2})
The following shows how each equation is multiplied both horizontally and vertically.
Equation (a) done horizontally:
Equation (a) done vertically:
Equation (b) done horizontally:
Equation (b) done vertically:
Equation (c) done horizontally:
Equation (c) done vertically: