Applying Exponent Rules: Scientific Notation 5 Section 6 Applying Exponent Rules: Scientific Notation

Section 5.6 Objectives 1 Convert Decimal Notation to Scientific Notation 2 Convert Scientific Notation to Decimal Notation 3 Use Scientific Notation to Multiply and Divide

Scientific Notation When a number has been written as the product of a number x, where 1  x < 10, and a power of 10, it is said to be in scientific notation. That is, a number is written in scientific notation when it is of the form x × 10N where 1  x < 10 and N is an integer. 8200 = 8.2  1000 = 8.2  103 Greater than 1 and less than 10 Power of 10 34,200,000 = 3.42  10000000 = 3.42  107 Scientific notation

Decimal Notation to Scientific Notation Steps to Convert from Decimal Notation to Scientific Notation To change a positive number into scientific notation: Step 1: Count the number N of decimal places that the decimal point must be moved in order to arrive at a number x, where 1  x < 10. Step 2: If the original number is greater than or equal to 1, the scientific notation is x × 10N. If the original number is between 0 and 1, the scientific notation is x × 10–N

Decimal Notation to Scientific Notation Example: Write 67,300 in scientific notation. What power? 67,300. = 6.73  10 Starting position of decimal point Ending position of decimal point The decimal point was moved 4 places to the left, so we use a power of 4. 67,300 = 6.73  104 A number that is larger than 10 and written in scientific notation will always have a positive exponent as the power of 10.

Decimal Notation to Scientific Notation Example: Write 0.048 in scientific notation. What power? 0.048 = 4.8  10 Starting position of decimal point Ending position of decimal point The decimal point was moved 2 places to the right, so we use a power of –2. 0.048 = 4.8  10–2 A number that is smaller than 1 and written in scientific notation will always have a negative exponent as the power of 10.

Scientific Notation to Decimal Notation Steps to Convert from Scientific Notation to Decimal Notation Step 1: Determine the exponent on the number 10. Step 2: If the exponent is positive, then move the decimal N places to the right. If the exponent is negative, then move the decimal |N| places to the left. Add zeros, as needed.

Scientific Notation to Decimal Notation Example: Write 9.1  104 in decimal notation. 9.1  104 = 9.1000  104 = 91,000 Move the decimal point 4 places to the right. Example: Write 6.72  10–3 in decimal notation. 6.72  10–3 = 6.72  10–3 = 0.00672 Move the decimal point 3 places to the left.

Multiplying Using Scientific Notation Example: Multiply: (3  10– 4) · (8  10– 5) Express the answer in scientific notation. (3  10– 4) · (8  10– 5) = (3 · 8)  (10– 4 · 10– 5) = 24  10– 9 Use the Product Rule. = (2.4  101)  10– 9 Convert 24 to scientific notation. = 2.4  10– 8 Use the Product Rule.

Dividing Using Scientific Notation Example: Divide: Express the answer in scientific notation. Use the Quotient Rule.