Review of Numbers

Greatest Common DivisorExponents

Sections 1.2 & 1.3

Previous Information

Types of numbers Natural

Whole Integers Rational Irrational

Purpose

In later mathematics, you will want to work with numbers in simplest form.

It is easier to work with small numbers rather than large numbers.

Prime Factorization

Breaking a number down into it’s combination of prime factors

To start prime factorization, ask yourself…“Will 2 divide this number…,Will 3 divide this number…,Will 5 divide this number…”

53

Factors of 15

3 5

15

3 5

3 51 1

Factor Tree

15

3 51 1

Alternate Method

153

Factors of 45

3 5 45

3 15

3 52 1

Factor Tree

45

3 52 1

Alternate Method

3 53

35

Greatest Common Divisor

Largest factor of 2 or more numbers

When finding – use prime factorization

Greatest Common Divisor of 15 & 45

3 5 45

3 15

3 52 1

3 53

15

3 5

3 51 1

Maximum number of common factors in each final row is one three and one 5

Therefore, the GCD is 3*5 = 15

GCD is 15

Greatest Common Divisor of 8 & 27

Note: there is

no Common Divisor,

8

2 4

23

2 22

27

3 9

33

3 33

What number divides all numbers evenly?

1

Therefore,The GCD =1

TEST YOURSELF

1. 12 & 182. 12 & 243. 18 & 324. 32 & 645. 12 & 546. 18 & 54

7. 18 & 648. 32 & 549. 24 & 6410.54 & 7211.14 & 15

Find the GCD for the following pairs of numbers

612232618

228181

Exponents

Repeated Multiplication be

So you would multiply b e times b*b*b*b*b*…*b {e times}

b = base e = exponent

Examples

24

72

x3

=2*2*2*2

=7*7

=x*x*x

=16

=49

= x3

Rules of Exponents: MULTIPLICATION

Multiply like Bases

am * an

32 * 34

Add exponents

am+n

32+4 = 36

Rules of Exponents: Exponents

Exp raised to an Exp

(am )n

(32)4

Multiply exponents

am*n

32*4= 38

Rules of Exponents: DIVISION

Divide like Bases

am

an

34 32

Subtract exponents

am-n

34-2 = 32

Rules of Exponents:

Qty raised to an Exp

(ab)m

(3x)4

Distribute exponents

ambm

34x4

Quantity to an Exponent

Rules of Exponents: Negative Exp

Number raised to a neg Exp

a-m

3-2

= the reciprocal

1am

1 132 9=

TEST YOURSELF: Rules of Exponents

23 * 24

54/53

(2/5)2

-32

(-3)2

(6x)3

6x3

7(-2)

23+4 = 27 = 128 54-3 = 51 = 5 22/52 = 4/25 -32 = -9 (-1)2 * 32

63x3 = 216x3

6x3

1/72 = 1/49

Terminology

Factor Prime number Composite number Prime Factorization Exponential Form Greatest Common

Divisor

Expanded Form Sum Difference Product Quantity Reciprocal

Conclusion

Any two are more numbers have a Greatest Common Divisor(either 1 or some other Factor)

To find the GCD use Prime Factorization

(match factors using the smallest exponential value)

Rules of Exponents

Assignment

On-line 12 GCD & Exponents