1. Be able to divide polynomials

2. Be able to simplify expressions involving powers of monomials by applying the division properties of powers.

xy

yx

2

4 33 5

3

23

yx

xy6

36 228 ab

ab

ba

Monomial: A number, a variable, or the product of a number

and one or more variables Constant: A monomial that is a real number.

Power: An expression in the form xn.

Base: In an expression of the form xn, the base is x.

Exponent: In an expression of the form xn, the exponent is n.

Quotient: The number resulting by the division of one number by another.

Repeated multiplication can be represented using exponents.

zyxzyyxxxx 2433

To expand a power, use the exponent to determine the number of times a base is multiplied by itself.

xxxx 888 32

Product of Powers: When two numbers with the same base are multiplied together, add the exponents and leave the base unchanged.

nmnm aaa Power of a Product: In a product raised to a power, the exponent applies to each factor of the product.

2222 33 yxxy

Power of a Power: When a power is raised to another power, multiply the exponents and leave the base unchanged.

nmnm aa

Remember: Follow the order of operations when applying more than one property!

3323 yxx 3323 yx 33533 yx 33527 yx 31527 yx

Simplify:2

33

x

x

Step 1: Rewrite the expression in expanded form

xx

xxx

x

x

33

2

3

Step 2: Simplify.

xx

xxx

3

For all real numbers a, and integers m and n:

nmn

m

aa

a Remember: A number divided by itself is 1.

233x x3

Simplify:

23

y

x

Step 1: Write the exponent in expanded form.

y

x

y

x

y

x 3332

Step 2: Multiply and simplify.

y

x

y

x 33

For all real numbers a and b, and integer m:

m

mm

b

a

b

a

yy

xx 33 2

223

y

x2

29

y

x

3

6

3

3

363

33

24

x

2

24

x

2

16

x

3

2

43

yx

x

3243

y

x

3

3233

y

x

3

627

y

x

Apply quotient of powers. Apply power of

a quotient.

Apply quotient of powers

Apply power of a quotient

Simplify

3

3233

y

x

Apply power of a power

1.

2.

233 2

2

4

xyxy

yx

322

a

ba

322

a

ba

312

32

22

baa

ba

32ab

3332 ba

338 ba

233 2

2

4

xyxy

yx

22

21313

233 2

2

42

2

4

yxyx

xyxy

yx

22

22 42

yxyx

22

22 42

yx

yx

222224 yx

8

THINK!

x2-2 = x0 = 1