Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Exponent Study Guide Laws of Exponents and Exponent Facts The Basics Exponential Notation β Repeated MULTIPLICATION Example: 3 × 3 × 3 × 3 × 3 = 35 An Exponent is a number that shows you how many times the number is to be used in multiplication. It is written as a small number to the right and above the base number. When the base is either a FRACTION or a NEGATIVE, you must use PARENTHESES. Multiplication of Exponents β Product Law **The bases must be the SAME** LAW xm β xn = x m + n When you Multiply numbers with exponents, you keep the common base and ADD the EXPONENTS Example: 35 × 37 = (3 × 3 × 3 × 3 × 3) × (3 × 3 × 3 × 3 × 3 × 3 × 3) =3×3×3 ×3×3×3×3×3 ×3×3×3×3 = 3 5 + 7 = 312 Power Raised to a Power LAW (xm)n = xm β n When you have a POWER RAISED TO A POWER, you MULTIPLY the EXPONENTS Example: (53)4 = (53)(53 )(53)(53) = (5 × 5 × 5) × (5 × 5 × 5) × (5 × 5 × 5) × (5 × 5 × 5) =5×5×5 ×5×5×5×5×5 ×5×5×5×5 = 512 = 53 × 4 Negative Exponents LAW x-n = 1 π₯π When you have a number with a NEGATIVE EXPONENT, you DROP THE BASE AND THE EXPONET DOWN TO THE DENOMINATOR AND MAKE THE EXPONENT POSITIVE Expanded Form of the Decimal in Exponential Notation 629.125 600 + 20 + 9 + .1 + .02 + .005 6 (100) + 2 (10) + 9 (1) + 1 (.1) + 2 (.01) + 5 (.001) (6 × 102) + (2 × 101) + (9 × 100) + ( 1 × 10 -1) + (2 × 10-2) + (5 × 10-3) COUNT YOUR ZEROS β That will give you your exponent FIND THE ONES PLACEβ That will always be 100 because anything raised to the zero power is 1 Product of Numbers with Exponents Positive Base β The Product will ALWAYS be POSITIVE Negative Base β ONLY if the base is NEGATIVE Use the Even/Odd Exponent Rule Negative Base/Even Exponent β Positive Product Negative Base/Odd Exponent β Negative Product Example: Does (β3)7 have a positive or negative product? Is the Base Positive or Negative? β Negative Even or Odd? β Odd NEGATIVE Division of Exponents β Quotient Law LAW **The bases must be the SAME** π₯π π₯π = x mβn When you divide numbers with exponents, you keep the common base and SUBTRACT the EXPONENTS ππ Example: π π = 4 × 4 × 4 × 4 × 4 × 4 × 4 × 4 × 4 = 45 4×4×4×4 ×4 Power Raised to a Power (FRACTION) (xy)n = xnyn π₯ π (π¦) = LAW π₯π π¦π When you have a QUANTITY RAISED TO A POWER, you must RAISE EACH TERM IN THAT QUANTITY TO THAT POWER (xm β yp)n = (xm)n β (yp)n π₯π π ( π¦π ) = (π₯π )π (π¦π )π Facts ANYTHING raised to the zero power is equal to 1 74850 = 1 (a3b 5c15)0 = 1 ANYTHING raised to the 1 power is equal to itself 74851 = 7485 (a 3b5c15)1 = a3b 5c15 Multiplication is COMMUTATIVE β This means you can CHANGE THE ORDER β Group your numbers together and your variables together when multiplying When simplifying numbers in exponential notation, all exponents should be positive and in one fraction (if applicable) Exponent Study Guide APPLICATION OF THE EXPONENT LAWS 3 7 Example 1) (2x ) (17x ) The operation between each term is multiplication 2 · x3 · 17 · x7 Multiplication is commutative, therefore, you can change the order 2 · 17 · x3 · x7 Use product law to simplify 34 · x3 + 7 = 34x10 Example 3) 35 27 Must have the same base in order to use the division law. It is easier to make the bigger number smaller rather than the smaller number bigger. 27 must have a base of 3 3 1 3 =3=3 32 = 3 × 3 = 9 33 = 3 × 3 × 3 = ππ Replace 27 with the equivalent 33 35 33 Use quotient law 35β3 = 32 Example 5) (π₯ 4 π¦π§ 5 )2 This really means (π₯ 4 π¦1 π§ 5 )(π₯ 4 π¦1 π§ 5 ) Since multiplication is commutative, we can change the order π₯ 4 π₯ 4π¦ 1π¦ 1π§ 5 π§ 5 We can then write this as (π₯ 4 )2 (π¦ 1)2 (π§ 5 )2 Use power to a power law π₯ 4 × 2π¦ 1 × 2π§ 5 × 2 = π± π π²π π³ππ Example 7) π₯ 3 π¦ β8 Since the two terms do not have the same base, you CANNOT use the product law. However, you can use the negative exponent law and make it in to one fraction. π₯3 1 π±π × 8= π 1 π¦ π² Example 2) (11π₯ 16 ) 6 π₯4 Put (11π₯ 16 ) over 1 in order to multiply two fractions together (11π₯16 ) 1 6 · 4 π₯ Multiply the numerators together, multiply the denominators together 6 (11π₯ 16 ) 6 β 11 β π₯ 16 66π₯ 16 = = π₯4 π₯4 π₯4 Use quotient law 66 β π₯ 16β4 = πππππ Example 4) 162 β 26 Must have the same base in order to use the multiplication law. It is easier to make the bigger number smaller rather than the smaller number bigger. 162 must have a base of 2 (16 )2 21 = 2 = 2 2 2 =2 ×2=4 23 = 2 × 2 × 2 = 8 24 = 2 × 2 × 2 × 2 = ππ Replace 16 with the equivalent 2 4 (24)2 β 26 Use power to a power law 2 4 × 2 β 26 = 28 β 26 Use product law 2 8 + 6 = 2 14 Example 6) π₯ 5 π₯ β6 Use the product law π₯ 5+(β6) Integer Rules β Addition β Same Signs Add, Same Signs Keep the Signs; Different Signs Subtract, Different Signs Keep the Sign of the Larger Absolute Value β Subtraction β Keep Change Change then Use Addition Rules π₯ β1 Use negative exponent law β drop it to the denominator and make the exponent positive 1 π = 1 π₯ π± π₯3 π¦7 Example 8) π₯12 Since the operation between x and y is multiplication, you can separate the fractions π₯3 π¦7 × π₯ 12 1 Use the quotient law (and Integer Rules!) π₯ 3β12 π¦7 = π₯ β9π¦ 7 Positive Exponents ONLY (Negative Exponent Law) 1 π¦7 π²π × = π₯9 1 π±π