Section 1.3 Integer Exponents

Objectives Bases and Positive Exponents Zero and Negative Exponents Product, Quotient, and Power Rules Order of Operations Scientific Notation

Bases and Positive Exponents The expression 82 is an exponential expression with base 8 and exponent 2. Exponent Base

Example Using the given base, write each number as an exponential expression. a. 100,000 (base 10) b. 128 (base 2) Solution a. 100,000 b. 128

INTEGER EXPONENTS Let a and b be nonzero real numbers and m and n be positive integers. Then 1. 2. 3. 4. 5.

Example Simplify each expression. a. b. c. d. Solution a. b. c. d.

For any number a and integers m and n, THE PRODUCT RULE For any number a and integers m and n,

Example Multiply and simplify. Use positive exponents. a. b. c. d. Solution a. b. c. d.

For any nonzero number a and integers m and n, THE QUOTIENT RULE For any nonzero number a and integers m and n,

Example Simplify each expression. Use positive exponents. a. b. c. d. Solution a. b. c. d.

For any real number a and integers m and n, RAISING POWERS TO POWERS For any real number a and integers m and n,

Example Simplify each expression. Use positive exponents. a. b. c. d.

For any real numbers a and b and integer n, RAISING PRODUCTS TO POWERS For any real numbers a and b and integer n,

Example Simplify each expression. Use positive exponents. a. b. c. d.

For nonzero numbers a and b and any integer n, RAISING QUOTIENTS TO POWERS For nonzero numbers a and b and any integer n,

Example Simplify each expression. Use positive exponents. a. b. c. d.

ORDER OF OPERATIONS Using the following order of operations, first perform all calculations within parentheses and absolute values, or above and below the fraction bar. Then use the same order of operations to perform the remaining calculations. 1. Evaluate all exponential expressions. Do any negations after evaluating exponents. 2. Do all multiplication and division from left to right. 3. Do all addition and subtraction from left to right.

Example Evaluate each expression. a. b.

SCIENTIFIC NOTATION A real number a is in scientific notation when a is written as b  10n , where 1 ≤ |b| < 10 and n is an integer.

3. If the decimal point was moved to the left, then a = b  10n. WRITING A POSITIVE NUMBER IN SCIENTIFIC NOTATION 1. Move the decimal point in a number a until it represents a number b such that 1 ≤ b < 10. 2. Count the number of decimal places that the decimal point was moved. Let this positive integer be n. (If the decimal point is not moved, then a = a  100.) 3. If the decimal point was moved to the left, then a = b  10n. If the decimal point was moved to the right, then a = b  10-n.

Important Powers of 10 Number 10-3 10-2 10-1 103 106 109 1012 Value Thousandth Hundredth Tenth Thousand Million Billion Trillion

Example Write each number in scientific notation. a. 475,000 b. 0.00000325 Solution a. 475,000 b. Move the decimal point 6 places to the right. Move the decimal point 5 places to the left.

Example Write each number in standard form. a. b. Solution Move the decimal point 6 places to the right since the exponent is positive. Move the decimal point 3 places to the left since the exponent is negative.