Section 5.6 Rules of Exponents and Scientific Notation

What You Will Learn Exponents Rules of Exponents Scientific Notation

Exponents When a number is written with an exponent, there are two parts to the expression: baseexponent The exponent tells how many times the base should be multiplied together.

Exponents 52, 2 is the exponent, 5 is the base Read 52 as “5 to the second power” or “5 squared,” which means

Exponents “5 to the third power,” or “5 cubed” is “b to the nth power,” or bn, means

Example 1: Evaluating the Power of a Number Evaluate. a) 52 = 5 • 5 = 25 b) (–3)2 = (–3) • (–3) = 9 c) 34 = 3 • 3 • 3 • 3 = 81 d) 11000 = 1 e) 10001 = 1000

Example 2: The Importance of Parentheses Evaluate. a) (–2)4 = (–2)(–2)(–2)(–2) = 4(–2)(–2) =–8(–2) = 16 b) –24 = –1 • 24 = –1 • 2 • 2 • 2 • 2 = –1 • 16 = –16

Example 2: The Importance of Parentheses Evaluate. c) (–2)5 = (–2)(–2)(–2)(–2)(–2) = 4(–2)(–2)(–2) =–8(–2)(–2) = 16(–2) = –32 b) –25 = –1 • 25 = –1 • 2 • 2 • 2 • 2 • 2 = –1 • 32 = –32

The Importance of Parentheses (–x)n ≠ –xn where n is an even natural number

Product Rule for Exponents

Example 3: Using the Product Rule for Exponents Use the product rule to simplify. a) 33 • 32 = 33+2 = 35 =243 b) 5 • 53 = 51 • 53 = 51+3 = 54 = 625

Quotient Rule for Exponents

Example 4: Using the Quotient Rule for Exponents Use the quotient rule to simplify. = 37–5 = 32 =9 = 59–5 = 54 = 625

Zero Exponent Rule

Example 5: The Zero Power Use the zero exponent rule to simplify. Assume x ≠ 0. a) 20 = 1 b) (–2)0 = 1 c) –20 = –1 • 20 = –1 • 1 = –1 d) (5x)0 = 1 e) 5x0 = 5 • x0 = 5 • 1 = 5

Negative Exponent Rule

Example 6: Using the Negative Exponent Rule Use the negative exponent rule to simplify.

Power Rule for Exponents

Example 7: Evaluating a Power Raised to Another Power Use the power rule to simplify. a) (54)3 = 54•3 = 512 b) (72)5 = 72•5 = 710

Rules for Exponents Product Rule Quotient Rule Zero Exponent Rule Negative Exponent Rule Power Rule

Scientific Notation Many scientific problems deal with very large or very small numbers. Distance from the Earth to the sun is 93,000,000,000,000 miles. Wavelength of a yellow color of light is 0.0000006 meter.

Scientific Notation Scientific notation is a shorthand method used to write these numbers. 93,000,000 = 9.3 × 10,000,000 = 9.3 × 107 0.0000006 = 6.0 × 0.0000001 = 6.0 × 10–7

To Write a Number in Scientific Notation 1. Move the decimal point in the original number to the right or left until you obtain a number greater than or equal to 1 and less than 10. 2. Count the number of places you have moved the decimal point to obtain the number in Step 1. If the decimal point was moved to the left, the count is to be considered positive.

To Write a Number in Scientific Notation If the decimal point was moved to the right, the count is to be considered negative. 3. Multiply the number obtained in Step 1 by 10 raised to the count found in Step 2. (The count found in Step 2 is the exponent on the base 10.)

Example 8: Converting from Decimal Notation to Scientific Notation Write each number in scientific notation. a) In 2010, the population of the United State was about 309,500,000. 309,500,000 = 3.095 × 108 b) In 2010, the population of the China was about 1,348,000,000. 1,348,000,000 = 1.348 × 109

Example 8: Converting from Decimal Notation to Scientific Notation c) In 2010, the population of the world was about 6,828,000,000. 6,828,000,000 = 6.828 × 109 d) The diameter of a hydrogen atom nucleus is about 0.0000000000011 millimeter. 0.0000000000011 = 1.1 × 10–12

Example 8: Converting from Decimal Notation to Scientific Notation e) The wavelength of an x-ray is about 0.000000000492. 0.000000000492 = 4.92 × 10–10

To Change a Number in Scientific Notation to Decimal Notation 1. Observe the exponent on the 10. 2. a) If the exponent is positive, move the decimal point in the number to the right the same number of places as the exponent. Adding zeros to the number might be necessary.

To Change a Number in Scientific Notation to Decimal Notation b) If the exponent is negative, move the decimal point in the number to the left the same number of places as the exponent. Adding zeros might be necessary.

Example 9: Converting from Scientific Notation to Decimal Notation Write each number in decimal notation. a) The average distance from Mars to the sun is about 1.4 × 108 miles. 1.4 × 108 = 140,000,000 b) The half-life of uranium-235 is about 4.5 × 109 years. 4.5 × 109 = 4,500,000,000

Example 9: Converting from Scientific Notation to Decimal Notation c) The average grain size in siltstone is about 1.35 × 10–3 inch. 1.35 × 10–3 = 0.00135 d) A millimicron is a unit of measure used for very small distances. One millimicron is about 3.94 × 10–8 inch. 3.94 × 10–8 = 0.0000000394

Example 10: Multiplying Numbers in Scientific Notation Multiply (2.1 × 105)(9 × 10–3). Write the answer in scientific notation and in decimal notation. (2.1 × 105)(9 × 10–3) = (2.1 × 9)(105 × 10–3) = 18.9 × 102 = 1890

Example 11: Dividing Numbers in Scientific Notation Divide . Write the answer in scientific notation and in decimal notation.

Example 11: Dividing Numbers in Scientific Notation Solution