Number Theory and the Real Number System CHAPTER 5 Number Theory and the Real Number System

Exponents and Scientific Notation 5.6 Exponents and Scientific Notation

Objectives Use properties of exponents. Convert from scientific notation to decimal notation. Convert from decimal notation to scientific notation. Perform computations using scientific notation. Solve applied problems using scientific notation.

Properties of Exponents Property Meaning Examples The Product Rule bm · bn = bm + n When multiplying exponential expressions with the same base, add the exponents. Use this sum as the exponent of the common base. 96 · 912 = 96 + 12 = 918 The Power Rule (bm)n = bmn When an exponential expression is raised to a power, multiply the exponents. Place the product of the exponents on the base and remove the parentheses. (34)5 = 34·5 = 320 (53)8 = 53·8 = 524 The Quotient Rule When dividing exponential expressions with the same base, subtract the exponent in the denominator from the exponent in the numerator. Use this difference as the exponent of the common base.

The Zero Exponent Rule If b is any real number other than 0, b0 = 1.

Example: Using the Zero Exponent Rule Use the zero exponent rule to simplify: a. 70=1 b. c. (5)0 = 1 d. 50 = 1

The Negative Exponent Rule If b is any real number other than 0 and m is a natural number,

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

Powers of Ten A positive exponent tells how many zeros follow the 1. For example, 109, is a 1 followed by 9 zeros: 1,000,000,000. A negative exponent tells how many places there are to the right of the decimal point. For example, 10-9 has nine places to the right of the decimal point. 10-9 = 0.000000001

Scientific Notation A positive number is written in scientific notation when it is expressed in the form a  10n , where a is a number greater than or equal to 1 and less than 10 (1 ≤ a < 10), and n is an integer.

Convert Scientific Notation to Decimal Notation If n is positive, move the decimal point in a to the right n places. If n is negative, move the decimal point in a to the left |n| places.

Example: Converting from Scientific to Decimal Notation Write each number in decimal notation: a. 1.375  1010 b. 1.1  10-4 In each case, we use the exponent on the 10 to move the decimal point. In part (a), the exponent is positive, so we move the decimal point to the right. In part (b), the exponent is negative, so we move the decimal point to the left.

Converting From Decimal to Scientific Notation To write the number in the form a  10n: Determine a, the numerical factor. Move the decimal point in the given number to obtain a number greater than or equal to 1 and less than 10. Determine n, the exponent on 10n. The absolute value of n is the number of places the decimal point was moved. The exponent n is positive if the given number is greater than or equal to 10 and negative if the given number is between 0 and 1.

Example: Converting from Decimal Notation to Scientific Notation Write each number in scientific notation: a. 4,600,000 b. 0.000023 Solution:

Computations with Scientific Notation We use the product rule for exponents to multiply numbers in scientific notation: (a 10n)  (b  10m) = (a  b)  10n+m Add the exponents on 10 and multiply the other parts of the numbers separately.

Example: Multiplying Numbers in Scientific Notation Multiply: (3.4  109)(2  10-5). Write the product in decimal notation. Solution: (3.4  109)(2  10-5) = (3.4  2)(109  10-5) = 6.8  109+(-5) = 6.8  104 = 68,000

Computations with Scientific Notation We use the quotient rule for exponents to divide numbers in scientific notation: Subtract the exponents on 10 and divide the other parts of the numbers separately.

Example: Dividing Numbers In Scientific Notation Divide: . Write the quotient in decimal notation. Solution: Regroup factors. Subtract the exponents. Write the quotient in decimal notation.

Example: The National Debt As of December 2011, the national debt was $15.2 trillion, or 15.2  1012 dollars. At that time, the U.S. population was approximately 312,000,000, or 3.12  108. If the national debt was evenly divided among every individual in the United States, how much would each citizen have to pay? Solution: The amount each citizen would have to pay is the total debt, 15.2  1012, divided among the number of citizens, 3.12  108.

Example: The National Debt continued Every citizen would have to pay approximately $48,700 to the federal government to pay off the national debt.