Scientific NotationMultiplication, 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 = 1012+4 = 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 = 10-5+3 = 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 = 10-4-0 = 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 5-2 = 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 = ?

• 8 - 5 = 3. The smaller exponent must be increased by 3.• 2.456×105 = 0.002456×108

• 0.002456×108 +6.0034×108 = 6.005856×108

• 6.005856×108 is in scientific notation.• Thus, 2.456×105 +6.0034×108 = 6.005856×108

• Example 2: 3.5×1018 + 5.3×1012 = ?

• 18 - 12 = 6. The smaller exponent must be increased by 6.• 5.3×1012 = 0.0000053×1018• 3.5×1018 + 0.0000053×1018 = 3.5000053×1018• 3.5000053×1018 is in scientific notation.• Thus, 3.5×1018 + 5.3×1012 = 3.5000053×1018 .

Subtraction

• Example 3: 5.10802×103 -6.1×10-2 = ?

• 3 - (-2) = 5. The smaller exponent must be increased by 5.• 6.1×10-2 = 0.000061×103

• 5.10802×103 -0.000061×103 = 5.107959×103

• 5.107959×103 is in scientific notation.• Thus, 5.10802×103 -6.1×10-2 = 5.107959×103 .

Subtractions (cont..)

• Example 4: 4.801×103 -2.2×107 = ?

• 7 - 3 = 4. The smaller exponent must be increased by 4.• 4.801×103 = 0.0004801×107

• 0.0004801×107 -2.2×107 = - 2.1995199×107

• -2.1995199×107 is in scientific notation.• Thus, 4.801×103 -2.2×107 = - 2.1995199×107 .

Subtractions (Cont..)

• Example 5. 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 measurednumber are significant. Number of Significant Figures

• 38.15 cm 4• 5.6 ft 2 • 65.6 lb 3 • 122.55 m 5

Leading zeros

• RULE 3. Leading zeros in decimal numbers are NOT significant.

Number of Significant Figures0.008 mm 1• 0.0156 oz 3• 0.0042 lb 2• 0.000262 mL 3

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. 2• 200. yr 3 if followed by a decimal, shows

measurements implies significant• 48,600 gal 3• 25,005,000 g 5

Trailing zeros

• RULE 5.Trailing zeroes in numbers with decimals ARE significant if theyare to the right of the decimal and a nonzero number. Number of Significant Figures• 16.000 cm 5• 160.0 min 4• 0.030 mL 2• 23.070 g 5

• State the number of significant figures in each ofthe following:

• A. 0.030 m 2• B. 4.050 L 4• C. 0.0008 g 1• D. 3.00 m 3• E.2,080,000 bees 3

Scientific notation

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

• How many significant figures are in each of the following measurements?• 2.4 x 101mL 2 significant figures • 3.001 x 103 g 4 significant figures• 3.20 x 10-2 m3 3• 6.4 x 10 4 molecules 2• 5.6 x 10 kg 2

Summary significant figures

Any digit that is not zero is significant1.234 kg 4 significant figures• Zeros between significant digits are significant606 m 3 significant figures• Zeros to the left of the first nonzero digit are not significant 0.08 L 1 significant figure• If a number is greater than 1, then all zeros to the right of the decimal point are significant 2.0 mg 2 significant figures• If a number is less than 1, then only the zeros that are at theend and in the middle of the number are significant0.00420 g 3 significant figures