ObjectivesThe student will be able to:

1. multiply monomials.

2. simplify expressions with monomials.

A monomial is a1. number,

2. variable, or

3. a product of one or more numbers and variables.

Examples:

5

y

3x2y3

Why are the following not monomials?

x + yaddition

division

2 - 3a

Subtraction

More than 1 term!

x

y

Rule 1. Multiplying MonomialsWhen multiplying monomials, you

ADD the exponents with same variables and multiply coefficients.

a) x2 • x4

x2+4 Base = x Exponent =

x6

b) (5x7 )(x6 )5x13

Rule 1. Multiplying MonomialsWhen multiplying monomials, you

ADD the exponents with same variables.

c) x • x (there is no exponent, what do you put?)

d) (5x7 )(5x7 )(5x7)

125x21

Example1. Simplify m3(m4)(m)

A. m7

B. m8

C. m12

D. m13

• Add exponents • Remember you are

adding 3 exponents, if an exponent isn’t given, always put a 1 there.

Example1b. Simplify 4m3(3m)(2m)

A. 24m5

B. 24m6

C. 24m4

D. 24m3

• Add exponents • Remember you are

adding 3 exponents, if an exponent isn’t given, always put a 1 there.

Ex2: More examples, with negatives

A. (-12abc)(4a2b4) *(only 1 c)*

B. (-a2b4) (1/4a3b5 )

C. (-3a4d) (a5d2)

D. z5 • ½z2

E. -5z5 •-z2

Example 3: You try

Multiply coefficients and ADD exponents with same variable!

a) (4ab6) (-7a2b3) b) (r4)(-12r7) c) (6cd5)(5c5d2) d) 2a2y3 • 3a3y4

Rule 2. Power to a PowerWhen you have an exponent with an

exponent, you multiply those exponents and multiply coefficients.

a) (x2)3

x2• 3

x6

b) (y3)4

y12

Rule 2. Power to a PowerWhen you have an exponent with an

exponent, you multiply those exponents and multiply coefficients.

c) [(32 ) 3] 2

d) [(23 ) 3] 2

Check with the calculator

Example 1. Simplify (p2)4

A. p2

D. p4

C. p8

D. p16

• Multiply exponents!

Rule 3. Power of a ProductWhen you have a power outside of the

parentheses, everything in the parentheses is raised to that power.

a) (2a)3 DISTRIBUTE (2 arrows)

23a3

8a3

b) (3x)2 (2 arrows)

9x2

Rule 3. Power of a ProductWhen you have a power outside of the

parentheses, everything in the parentheses is raised to that power.

c) (2ab)3 DISTRIBUTE (3 arrows)

23a3b3

8a3b3

d) (3x4v)2 (4 arrows)

9x216v2

Example 5. Simplify (4r)3

A. 12r3

B. 12r4

C. 64r3

D. 64r4

• Draw 2 arrows and DISTRIBUTE.

Example 1, You Try:

a) (p1)2

b) (3p2)4

c) (5xy2)4

MULTIPLY

DISTRIBUTE-2 arrows

DISTRUBTE-3 arrows

Rule 4. Simplify (all rules)This is a combination of all of the rules.

a) (x3y2)4

x3• 4 y2• 4

x12 y8

b) (4x4y3)3

64x12y9

C) (2 v3w4)(3vw3)2

Example 6. Simplify (3a2b3)4

A. 12a8b12

B. 81a6b7

C. 81a16b81

D. 81a8b12

• Draw arrows and DISTRIBUTE!

Combining terms

• (-8 v3w4)(-3vw3)2

• (-5 v7w4)2 (v3w3)2

All Rules Combined:

1. x(x4)(x6) =

2. (4a4b4 )(9 a2b3) =

3. [(23)2]3 =

4. (3y5z)2 =

5. (-4mn2) (12m2n) =

6. (-2 v3w4)(-3vw3)2 =

Warm-UP, What to do with the exponents? Add? multiply?• 1. y(y6) =

• 2. (3p2)4 =

• 3. (2a2)(8a) =

• 4. (rs)(rs3)(s2) =

• 5. (-4x3) (-5x7) =

• 6. (-2 v3w4)(-3vw3)2 =

• 7. [(23)2]3 =

Lets review the 4 rules:

1.When you multiply powers with the same base you______________

2. When you have a power to a power (powers are side by side and stacked) you ________

3. When you have a power on the outside of the parenthesis you ___________________

4. Simplify/combine all the rules.

Practice a few more

1. (-10x3yz2)(-2xy5 ) =

2. (-x3)(-x4) =

3. (2 a2)(8a) =

Using Geometry

How do you find AREA =

How do you find VOLUME =

Example 1: Express the area of the square as a monomial.

What do you remember about a square?

Area = 4ab

Example 2: Find area

2n2

5n3

Area of a triangle:

4ab5

3a4b

Area of a triangle: