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Test Write the following in either standard form or scientific notation. 1. 0.000006824 2. 3.71 x 100 Simplify the following. Write your answer in both scientific notation and standard form. 3. 7.1 x 105 – 3.26 x 105 10/30/14 Exponential Form: 45 x 43 or 59 56 Expanded Form: 45 x 43 = (4x4x4x4x4)(4x4x4) 59 56 = 5∗5∗5∗5∗5∗5∗5∗5∗5 5∗5∗5∗5∗5∗5 Standard Form: 45 x 43 = 65,536 59 56 = 625 RULE: When multiplying two exponential expressions with the same base, you ADD the exponents. Example: 59 x 53 = 59+3 =512 The bases are the same (5), therefore you add the two exponents (9+3). RULE: When dividing two exponential expressions with the same base, you SUBTRACT the exponents. Example: 38 36 = 38-6 = 32 The bases are the same (3), therefore, you can subtract the exponents (8-6). When DIVIDING two exponential expressions that will result in an expression with a negative exponent, you have two options: 1. Divide by subtracting the exponents. 43 48 = _8 3 4 = _5 4 2. Write the numerator and denominator in expanded form and simplify. 43 48 = 4∗4∗4 4∗4∗4∗4∗4∗4∗4∗4 = 1 45 1 = 5 4 HINT: You must know your rules for operations with integers in order to be able to successfully solve problems with negative exponents! _ 45 Anything raised to the power of zero (0) is always one (1). 70 = 1 • Anything raised to the power of one (1) is always itself. 71 = 7 When an exponential expression is raised to a power, you multiply the two exponents. (88)6 = 88x6 =848 (1011)2 =1011x2 =1022 Addition (when the signs are the same) 1. Keep the Sign 2. Add (-9) + (-4) = Addition (when the signs are different) 1. Keep the sign of the number with the greater absolute value. 2. Subtract the bigger number from the smaller number. (-7) + 4 = Subtraction (Think L-C-O…Leave, Change, Opposite) 1. Change the subtraction sign to addition sign. 2. Change the sign of the second number. 3. Follow Addition Rules. Multiplication/Division 1. Positive & Positive = Positive 2. Negative & Negative = Positive 3. Positive & Negative = Negative 4. Negative & Positive = Negative 8 – 11 = 15 – (-7) = (-5) x (-9) = (-90) ÷ 3= Scientific Notation: a way to write a number as a product of the number, a, and 10n, when 1≤ a <10 (a needs to be at least equal to 1 but less than 10) and n is an integer. a x 10n 5.21 x 106 Standard Form: a way to write a number using a digit for each place. 591,157.21 5.12 x 106 Step 1: Simplify 106 106 = 10 x 10 x 10 x 10 x 10 x 10 = 1,000,000 Step 2: Multiply by 5.12 5.12 x 1,000,000 = 5,120,000 Hint: The exponent tells you how many spaces to move the decimal. When converting from scientific notation to standard form and the exponent is positive, you move the decimal to the RIGHT and fill spaces with zeroes. 860,000 Step 1: Step 2: Identify the location of the decimal point in 860,000. In all whole numbers, the decimal point is at the end (all the way to the right) of a number. 860,000. Decimal Point Move the decimal point to the left until you have a number that is greater than or equal to 1 and less than 10. Count the number of places you moved the decimal point. 860,000. the decimal is moved 5 places Step 3: Rewrite the number in scientific notation. The number of places you moved the decimal is the exponent for the base of 10. 8.6 x 105 Remember: Scientific Notation requires that the value for “a” be at least 1 and less than 10. Power of Zero 4.5 x 100 = 4.5 x 1 = 4.5 Negative Exponents Standard form to Scientific notation: 0.00000651 = 6.51 x 10-6 Count the number of paces the decimal is moved to the right to make the number between 1 and 10, the number of places moved to the right is written as the exponent for 10 and should be negative. Scientific Notation to Standard form: 8.75 x 10-7 = 0.000000875 Move the decimal to the left according to the number in the exponent. Ordering Use the values of the exponents to help determine the order. The smaller the exponent, the smaller the value. The greater the exponent, the greater the value. Write the number in standard form to check the order. Write the numbers in order in their original form (scientific notation). To estimate: 1. 2. Look for the greatest place value and round to that place value. Follow the rules for converting from standard form to scientific notation. Examples: Step 1: Step 2: Solution: 0.00005146 456,145,956 (2.15 x 108) x (1.24 x 103) Step 1: use the commutative and associative properties of multiplication to regroup and reorder the multiplication problem. Step 2: Multiply Step 1: (2.15 x 1.24) x (108 x 103) Step 2: 2.15 108+3 =1011 x1.24 Solution: 2.666 x 1011 Standard form: 266,600,000,000 16.4 x 109 4.1 x 105 Step 1: Divide the factors of a. a x 10n a a x 10n a Step 2: Then, apply the rules for dividing exponential expressions. (you subtract the exponents) Step 1: 16.4 = 4 4.1 Step 2: 109-5 =104 n-n Solution: Rewrite the answer in scientific notation using the number in step 1 and 2. 4 x 104 Standard form: 40,000 Examples: Step 1: To add or subtract in scientific notation, the exponents must be the same. If they are not the same, rewrite the terms so that the exponents are the same. To do so, determine the number by which to increase the smaller exponent by so it is equal to the larger exponent. Increase the smaller exponent by this number and move the decimal point of the number with the smaller exponent to the left the same number of places. 4.2 x 103 + 2.9 x 103 same exponents Step 2: Add or subtract the new factors (digits) for “a.” 4.2 + 2.9 = 7.1 Solution: Write the sum in scientific notation. If the answer is not in scientific notation (i.e. if “a” is not between 1 and10 ) convert it to scientific notation. 7.1 x 103 8.5 x 108 – 6.2 x 106 2.1 x 104 + 3.6 x 105