A **monomial** is an algebraic expression that consists of only one term. (A *term* is a numerical or literal expression with its own sign.) For instance, 9 *x*, 4 *a*^{2}, and 3 *mpx*^{2} are all monomials. The number in front of the variable is called the numerical coefficient. In 9 *x*, 9 is the coefficient.

Adding and subtracting monomials

To *add* or *subtract monomials,* follow the same rules as with signed numbers, *provided that the terms are alike.* Notice that you add or subtract the coefficients only and leave the variables the same.

Example 1

Perform the operation indicated.

- 3
*x*+ 2*x*= 5*x*

Remember that the rules for signed numbers apply to monomials as well.

Multiplying monomials

Reminder: The rules and definitions for powers and exponents also apply in algebra.

Similarly, *a* · *a* · *a* · *b* · *b* = *a*^{3} *b*^{2}.

To *multiply monomials,* add the exponents of the same bases.

Example 2

Multiply the following.

- (
*x*^{3})(*x*^{4}) =*x*^{3 + 4}=*x*^{7} - (
*x*^{2}*y*)(*x*^{3}*y*^{2}) = (*x*^{2}*x*^{3})(*yy*^{2}) =*x*^{2 + 3}*y*^{1 + 2}=*x*^{5}*y*^{3} - (6
*k*^{5})(5*k*^{2}) = (6 × 5)(*k*^{5}*k*^{2}) = 30*k*^{5 + 2}= 30*k*(multiply numbers)^{7} - –4(
*m*^{2}*n*)(–3*m*^{4}*n*^{3}) = [(–4)(–3)](*m*^{2}*m*^{4})(*nn*^{3}) = 12*m*^{2 + 4}*n*^{1 + 3}= 12*m*^{6}*n*^{4}(multiply numbers) - (
*c*^{2})(*c*^{3})(*c*^{4}) =*c*^{2 + 3 + 4}=*c*^{9} - (3
*a*^{2}*b*^{3}*c*)(*b*^{2}*c*^{2}*d*) = 3(*a*^{2})(*b*^{3}*b*^{2})(*cc*^{2})(*d*) = 3*a*^{2}*b*^{3 + 2}*c*^{1 + 2}*d*= 3*a*^{2}*b*^{5}*c*^{3}*d*

Note that in example (d) the product of –4 and –3 is +12, the product of *m*^{2} and *m*^{4} is *m*^{6}, and the product of *n* and *n*^{3} is *n*^{4}, because any monomial having no exponent indicated is assumed to have an exponent of l.

When monomials are being raised to a power, the answer is obtained by multiplying the exponents of each part of the monomial by the power to which it is being raised.

Example 3

Simplify.

- (
*a*^{7})^{3}=*a*^{21} - (
*x*^{3}*y*^{2})^{4}=*x*^{12}*y*^{8} - (2
*x*^{2}*y*^{3})^{3}= (2)^{3}*x*^{6}*y*^{9}= 8*x*^{6}*y*^{9}

Dividing monomials

To *divide monomials,* subtract the exponent of the divisor from the exponent of the dividend of the same base.

Example 4

Divide.

- 6.

You can simplify the numerator first.

Or, because is the numerator is all multiplication, you can reduce,

Working with negative exponents

Remember, if the exponent is negative, such as *x*^{–3}, then the variable and exponent may be dropped under the number 1 in a fraction to remove the negative sign as follows.

Example 5

Express the answers with positive exponents.