Greatest Common Divisor Exponents Sections 1.2 & 1.3 Review of Numbers Greatest Common Divisor Exponents Sections 1.2 & 1.3

Whole Previous Information Types of numbers Natural 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…”

Factors of 15 15 15 3 5 3 5 3 5 3 5 Alternate Method Factor Tree 3 5 1

Factors of 45 45 45 3 15 3 3 15 5 3 5 3 3 5 3 5 Alternate Method Factor Tree 3 5 45 45 3 15 3 3 15 5 2 1 3 5 3 3 5 2 1 3 5

Greatest Common Divisor Largest factor of 2 or more numbers When finding – use prime factorization

Greatest Common Divisor of 15 & 45 3 5 45 15 Maximum number of common factors in each final row is one three and one 5 3 15 3 5 Therefore, the GCD is 3*5 = 15 3 3 5 1 1 3 5 GCD is 15 2 1 3 5

Greatest Common Divisor of 8 & 27 Note: there is no Common Divisor, 8 27 Therefore, The GCD =1 2 4 3 9 What number divides all numbers evenly? 2 2 2 3 3 3 1 3 3 2 3

TEST YOURSELF Find the GCD for the following pairs of numbers 12 & 18 12 & 24 18 & 32 32 & 64 12 & 54 18 & 54 18 & 64 32 & 54 24 & 64 54 & 72 14 & 15 6 12 2 32 18 2 8 18 1

Exponents Repeated Multiplication be b = base e = exponent So you would multiply b  e times b*b*b*b*b*…*b {e times}

Examples 24 72 x3 =2*2*2*2 =7*7 =x*x*x =16 =49 = x3

Rules of Exponents: MULTIPLICATION Add exponents am+n 32+4 = 36 Multiply like Bases am * an 32 * 34

Rules of Exponents: Exponents Multiply exponents am*n 32*4= 38 Exp raised to an Exp (am )n (32)4

Rules of Exponents: DIVISION Subtract exponents am-n 34-2 = 32 Divide like Bases am an 34 32

Distribute exponents Qty raised to an Exp ambm (ab)m (3x)4 34x4 Rules of Exponents: Quantity to an Exponent Distribute exponents ambm 34x4 Qty raised to an Exp (ab)m (3x)4

Rules of Exponents: Negative Exp = the reciprocal 1 am 1 1 32 9 Number raised to a neg Exp a-m 3-2 =

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