A monomial is a number, a variable, or the product of a number and one or more variables. In other words, a monomial has one term.

A constant is a monomial that is a real number.

Examples of monomials:
5
2ab
^5x/₃
4x²y³

Examples that are not monomials:
⁵/_x is not a monomial because it is a quotient of a number and a variable and not a product

4a + 5b is not a monomial because it is a sum and has two terms

5x - 7y² is not a monomial because it is a difference and has two terms

Remember:  Terms are separated by plus or minus signs.

We need to be aware of our definition of monomial when talking about an expression like 5(2a + 3b).  5(2a + 3b) is a monomial because it is a product of 5 and (2a + 3b).

To develop our rules for working with exponents, we need to review the concept of power. Remember, in an expression like a^x, a is called the base, x is called the exponent, and a^x is called a power.

Also, remember that the exponent tells us how many times to multiply the base times itself.

Examples:
2³ = 2 · 2 · 2
5⁴ = 5 · 5 · 5 · 5

We can now expand our concept of power to develop our rule for multiplying exponents or product of powers.

Examples:

1.	22 · 23	Reasoning:
	2 · 2 · 2 · 2 · 2 25 or 22+3 = 25	Write in factored form Count the number of factors (2s) Looking at just the exponents, we can add 2 + 3 = 5
2.	24 · 23	Reasoning:
	2 · 2 · 2 · 2 · 2 · 2 · 2 27 or 24+3 = 27	Write in factored form Count the number of factors (2s) Looking at just the exponents, we can add 4 + 3 = 7

As you can see, writing our exponential form in factored form would become quite a pain if we had to write out 2²⁵ · 2¹⁵ in factored form. This is why it is important for us to know our rules in Algebra.

Rule for Multiplying Exponents or Product of Powers:

a^x · a^y = a^x+y
If the bases are the same, then we add the exponents.

Examples:

		Reasoning:
1.	m5 · m4 = m5 + 4 = m9	Identify the bases as being the same, m Add the exponents 5 + 4 = 9 Product of Powers:  You may add the exponents mentally. In other words, you would not have to write down 5 + 4.
2.	a3 · a2 · a4 = a3 + 2 + 4 = a9	Identify the bases as being the same, a Add the exponents 3 + 2 + 4 = 9 Product of Powers:  You may add the exponents mentally. In other words, you would not have to write down 3 + 2 + 4.

We can now extend our process for multiplying monomials to include signs and numbers as well as variables.

To multiply monomials:

1.	Signs	Use your rules for multiplication to find the sign of your answer first
2.	Numbers	Multiply the numbers
3.	Variables, add exponents	Multiply the variables by adding exponents

Review of rules for multiplication:

1. Two numbers
   1. Like signs, positive
   2. Unlike signs, negative
2. More than two numbers
   1. Count the minus signs
      1. Even, positive
      2. Odd, negative

Examples:

1.	-3x(2x)	Reasoning:
	-6x2	Unlike signs; negative Multiply the numbers, 3(2) = 6 Add the exponents, 1 + 1 = 2
Remember if an exponent is not written, then it is understood to be one.
2.	-2a2(-5a5)	Reasoning:
	10a7	Like signs, positive Multiply the numbers, 2(5) = 10 Add the exponents, 2 + 5 = 7
3.	-4x2y3(2x3y)(-7xy2)	Reasoning:
	56x6y6	Since we are multiplying 3 monomials, count the minus signs. There are two, which is an even number, so our answer is positive. Multiply the numbers, 4(2)(7) = 56 Add the exponents. Since there are two variables, we must do this twice. For the xs, 2 + 3 + 1 = 6 For the ys, 3 + 1 + 2 = 6
4.	-3a2b3c(-4a3b4c2)(-5a4b2c3)	Reasoning:
	-60a9b9c6	We are multiplying 3 monomials. Count the minus signs. There are three, which is an odd number, so our answer is negative. Multiply the numbers, 3(4)(5) = 60 Add the exponents on each variable: For the as, 2 + 3 + 4 = 9 For the bs, 3 + 4 + 2 = 9 For the cs, 1+ 2 + 3 = 6

We will now expand our rule for multiplying with exponents to include rules dealing with powers.

Power of a Power
(a^x)^y = a^xy
Reasoning: since parentheses tell us to multiply, we multiply the exponents.

Examples:

1.	(a3)2	Reasoning:
	a3 · a3 a3+3 a6	To develop the rule for power of a power: Write the expression in factored form (a3)2 = a3 · a3 Use our rule for multiplying, add exponents.
2.	(m3)4	Reasoning:
	m3 · m3 · m3 · m3 m3+3+3+3 m12	To develop the rule for power of a power: Write the expression in factored form (m3)4 = m3 · m3 · m3 · m3 Add exponents, 3 + 3 + 3 + 3 = 12

As you can see from these examples, it is better for us to know the rule for power of a power and to understand that the concept of parenthesis tells us to multiply.

There are two additional examples for taking powers:

Power of a Product	(ab)x = axbx
Power of a Power	(axby)z = axzbyz

Each power rule involves parenthesis, so the rules are all the same. Since parenthesis tell us to multiply, anytime you are working with a power involving exponents, you multiply the exponents.

Examples for Power of a Product:

1.	(xy)3	Reasoning:
	x3y3	Remember that the exponents on the x and y are one. Since parenthesis tell us to multiply: For the x, 1(3) = 3 For the y, 1(3) = 3
2.	(3mn)2	Reasoning:
	32m2n2 9m2n2	Multiply each exponent inside the ( ) by two. For the 3, 1(2) = 2 For the m, 1(2) = 2 For the n, 1(2) = 2 Evaluate the power, 32 = 3 · 3 = 9

Using the power of product rules is the same as using the distributive property. Now we are multiplying the exponent outside the ( ) by each exponent inside the ( ).

Examples for Power of a Monomial:

1.	(a2b3)4	Reasoning:
	a8b12	Multiply the exponent outside the ( ) by each exponent inside the ( ). For the a, 2(4) = 8 For the b, 3(4) = 12
2.	(-3mn4)2	Reasoning:
	32m2n8 9m2n8	Find the sign of your answer. Since two is even, the answer is positive. Multiply the exponent of 2 outside the ( ) by each exponent inside the ( ). For the 3, 1(2) = 2 For the m, 1(2) = 2 For the n, 4(2) = 8 Evaluate the power, 32 = 3 · 3 = 9
3.	(-2p4q5)3	Reasoning:
	-23p12q15 -8p12q15	Find the sign of your answer. Since 3 is odd, your answer is negative. Multiply the exponent of 3 outside the ( ) by each exponent inside the ( ). For the 2, 1(3) = 3 For the p, 4(3) = 12 For the q, 5(3) = 15 Evaluate the power, 23 = 2 · 2 · 2 = 8

The exponent will tell us the sign of our answer. If the exponent is even, our answer is positive. If the exponent is odd, our answer is negative.

We can also combine our different rules in the same problem. To do this, we must remember: in order of operations problems, we evaluate the powers before we multiply.

Examples:

1.	3x2y3(2x3y)2	Reasoning:
	3x2y3(22x6y2) 3x2y3(4x6y2) 12x8y5	Simplify the power by multiplying each exponent inside the ( ) by 2. For the 2, 1(2) = 2 For the x, 3(2) = 6 For the y, 1(2) = 2 Evaluate 22 = 2 · 2 = 4 Multiply the monomials by first doing the signs, then the numbers and finally the variables by adding exponents. Like signs, positive 3(4) = 12 For the xs, 2 + 6 = 8 For the ys, 3 + 2 = 5
2.	(-4m2n3)3(-2mn2)2	Reasoning:
	-43m6n9(22m2n4) -64m6n9(4m2n4) -256m8n13	Find the sign of each power: 3 is odd, so our first answer is negative: 2 is even so the second answer is positive. Multiply the exponents outside the ( ) by each exponent inside the ( ). For the 4, 1(3) = 3 For the m, 2(3) = 6 For the n, 3(3) = 9 For the 2, 1(2) = 2 For the m, 1(2) = 2 For the n, 2(2) = 4 In the first ( ), evaluate 43 = 4 · 4 · 4 = 64 In the second ( ), evaluate 22 = 2 · 2 = 4 Multiply the two monomials together. Unlike signs, negative Numbers 64(4) = 256 On the variables, add exponents. For the m, 6 + 2 = 8 For the n, 9 + 4 = 13

A review of our rules for working with exponents!

1.	Multiplication, add exponents	am · an = am+n
2.	Power of a Power, multiply exponents	(am)n = amn
3.	Power of a Product, multiply exponents	(ab)m = ambm
4.	Power of a Monomial, multiply exponents	(axby)m = amxbmy

The exponent tells us the sign of our answer:

To multiply, the bases must be the same.