# Multiplication, Division, Addition and Subtraction

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Multiplication, Division, Addition and Subtraction
Scientific Notation Multiplication, Division, Addition and Subtraction

Multiplication Quantities with exponents can be multiplied and divided easily if they have the same base. Since all number in scientific notation have base 10 , we can always multiply them and divide them. To multiply two numbers in scientific notation, multiply their coefficients and add their exponents.  The answer must be converted to scientific notation. Here are the steps to multiply two numbers in scientific notation: Multiply the coefficients--round to the number of significant figures in the coefficient with the smallest number of significant figures. Add the exponents. Convert the result to scientific notation.

Multiplication (Cont..)
Example 1:  (5.60×1012)×(7.102×104) = ? 5.6×7.102 = 39.8 1012×104 =  = 1016  (5.60×1012)×(7.102×104) = 39.8×1016 39.8×1016 = 3.98×1017 Thus, (5.6×1012)×(7.102×104) = 3.98×1017 .

Multiplication (Cont..)
Example 2:  (5.3201×10-5)×(1.8×103) = ? 5.3201×1.8 = 9.6 10-5×103 =  = 10-2  (5.3201×10-5)×(1.8×103) = 9.6×10-2 9.6×10-2 is in scientific notation. Thus, (5.3201×10-5)×(1.8×103) = 9.6×10-2 .

Division Here are the steps to divide two numbers in scientific notation: Divide the coefficients--round to the number of significant figures in the coefficient with the smallest number of significant figures. Subtract the exponents. Convert the result to scientific notation.

Division (cont..) Example 3: (4.14×10-4)÷(8.28×10 0) = ?
4.14/8.28 = 0.500 10x10-4/10x10 0 = = 10-4 (4.14×10-4)÷(8.28×10 0) = 0.500×10-4 0.500×10-4 = 5.00×10-5 Thus, (4.14×10-4)÷(8.28×100) = 5.00×10-5 .

Division (Cont..) Example 4: (3.04×10 5)÷(9.89×10 2) = ?
3.04/9.89 = 0.307 10x 5/10x 2 = = 10 3 (3.04×105)÷(9.89×102) = 0.307×10 3 0.307×10 3 = 3.07×10 2 Thus, (3.04×10 5)÷(9.89×10 2) = 3.07×10 2 .

Addition One of the properties of quantities with exponents is that numbers with exponents can be added and subtracted only when they have the same base and exponent. Since all numbers in scientific notation have the same base (10), we need only worry about the exponents. 5.1x10 To be added or subtracted, two numbers in scientific notation must be manipulated so that their bases have the same exponent--this will ensure that corresponding digits in their coefficients have the same place value.

Addition (cont..) Here are the steps to adding or subtracting numbers in scientific notation : 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. (i.e. divide by the appropriate power of 10 .) Add or subtract the new coefficients. If the answer is not in scientific notation (i.e. if the coefficient is not between 1and 10 ) convert it to scientific notation. The answer should include coefficient, base, and exponent. Note: If the numbers start with the same exponents, their coefficients can be added, but be careful--the answer might need to be converted to scientific notation.  .

Addition Example 1: 2.456×105 +6.0034×108 = ?
2.456×105  ×108 = ? 8 - 5 = 3. The smaller exponent must be increased by 3. 2.456×105 = ×108 ×108  ×108 = ×108 ×108 is in scientific notation. Thus, 2.456×105  ×108 = ×108

Example 2:  3.5× ×1012 = ? = 6. The smaller exponent must be increased by 6. 5.3×1012 = ×1018 3.5× ×1018 = ×1018 ×1018 is in scientific notation. Thus, 3.5× ×1012 = ×1018 .

Subtraction Example 3: 5.10802×103 -6.1×10-2 = ?
×103 -6.1×10-2 = ? 3 - (-2) = 5. The smaller exponent must be increased by 5. 6.1×10-2 = ×103 ×103  ×103 = ×103 ×103 is in scientific notation. Thus,  ×103 -6.1×10-2 = ×103 .

Subtractions (cont..) Example 4: 4.801×103 -2.2×107 = ?
4.801×103 -2.2×107 = ? 7 - 3 = 4. The smaller exponent must be increased by 4. 4.801×103 = ×107 ×107 -2.2×107 = ×107 ×107 is in scientific notation. Thus, 4.801×103 -2.2×107 = ×107 .

Subtractions (Cont..) Example 5. 1.4×10-5 -5.67×10-6 = ?
1.4×10-5 -5.67×10-6 = ? -5 - (-6) = 1. The smaller exponent must be increased by 1. 5.67×10-6 = 0.567×10-5 1.4×10-5 -0.567×10-5 = 0.833×10-5 0.833×10-5 = 8.33×10-6 in scientific notation. Thus, 1.4×10-5 -5.67×10-6 = 8.33×10-6 .

Significant Figures RULE 1. All non-zero digits in a measured
number are significant. Number of Significant Figures 38.15 cm 5.6 ft 65.6 lb m

Leading zeros RULE 3. Leading zeros in decimal numbers are NOT significant. Number of Significant Figures 0.008 mm oz lb mL

Trailing zeros RULE 4. Trailing zeros in numbers without decimals are NOT significant. They are only serving as place holders. Number of Significant Figures 25,000 in 200. yr if followed by a decimal, shows measurements implies significant 48,600 gal 25,005,000 g

Trailing zeros RULE 5. Trailing zeroes in numbers with decimals ARE significant if they are to the right of the decimal and a nonzero number. Number of Significant Figures cm 160.0 min 0.030 mL g

State the number of significant figures in each of
the following: A m B L C g D m E.2,080,000 bees

Scientific notation ALL numbers before the times sign are significant.
Number of Significant Figures 1.80 x 10 4 cm 2.000 x 10 3 min

How many significant figures are in each of the following measurements?
2.4 x 101mL significant figures 3.001 x 103 g significant figures 3.20 x 10-2 m 6.4 x 10 4 molecules 5.6 x 10 kg

Summary significant figures
Any digit that is not zero is significant 1.234 kg significant figures Zeros between significant digits are significant 606 m significant figures Zeros to the left of the first nonzero digit are not significant 0.08 L significant figure If a number is greater than 1, then all zeros to the right of the decimal point are significant 2.0 mg significant figures If a number is less than 1, then only the zeros that are at the end and in the middle of the number are significant g significant figures