1.2 Exponents and Scientific Notation
- 1. 1.2 Exponents and Scientific Notation Chapter 1 Prerequisites
- 2. Concepts & Objectives ⚫ Objectives for this section: ⚫ Use exponent rules. ⚫ Find the power of a product and a quotient. ⚫ Simplify exponential expressions. ⚫ Use scientific notation.
- 3. Properties of Exponents ⚫ Recall that for variables x and y and integers a and b: Product Rule Quotient Rule Power Rule Zero Exponent Negative Rule + = a b a b x x x − = a a b b x x x ( ) = b a ab x x − = 1 a a x x = 0 1 x
- 4. Simplifying Exponents ⚫ Example: Simplify ⚫ 1. ⚫ 2. ⚫ 3. − 3 2 2 2 5 25 5 x y z xy z ( ) − 4 2 3 2r s t ( ) 4 2 5 3x y − −
- 5. Simplifying Exponents ⚫ Example: Simplify ⚫ 1. ⚫ 2. ⚫ 3. − 3 2 2 2 5 25 5 x y z xy z ( ) − 4 2 3 2r s t ( ) 4 2 5 3x y − − 3 1 2 2 2 5 25 5 x y z − − − − = ( ) ( ) ( ) 4 2 4 3 4 1 4 2 r s t − = ( ) ( ) ( ) 4 2 4 5 4 3 x y − = − − − = 2 4 3 5x y z − = 8 12 4 16r s t 8 20 81x y − =
- 6. Scientific Notation ⚫ A shorthand method of writing very small and very large numbers is called scientific notation, in which we express numbers in terms of exponents of 10. ⚫ To write a number in scientific notation ⚫ Move the decimal point to the right of the first nonzero digit in the number. ⚫ Write the digits as a decimal number between 1 and 10. ⚫ Count the number of places n that you moved the decimal point. ⚫ Multiply the decimal number by 10 raised to a power of n. If you moved the decimal left (large number), n is positive; if you moved it right (small number), n is negative.
- 7. Scientific Notation (cont.) Example: Write 2,780,000 in scientific notation.
- 8. Scientific Notation (cont.) Example: Write 2,780,000 in scientific notation. ⚫ Moving the decimal 6 places to the left means n = 6, therefore our answer is ⚫ Notice that we can drop the extra zeros. 6 places to the left 6 2.78 10
- 9. Scientific Notation (cont.) Example: Write 0.00000000000047 in scientific notation.
- 10. Scientific Notation (cont.) Example: Write 0.00000000000047 in scientific notation. ⚫ Moving the decimal 13 places to the right means n = ‒13, therefore our answer is 13 places to the left 13 4.7 10−
- 11. Scientific Notation (cont.) ⚫ To convert a number in scientific notation to standard notation, simply reverse the process. ⚫ Move the decimal n places to the right if n is positive or n places to the left if n is negative. ⚫ Add zeros as needed. ⚫ Remember, if n is positive, the absolute value of the number is greater than 1, and if n is negative, the absolute value of the number is less than 1.
- 12. Scientific Notation (cont.) Example: Convert 3.547×1014 to standard notation. Shift the decimal point 14 places to the right and fill in the blanks with 0s. 354700000000000 or 354,700,000,000,000
- 13. Scientific Notation (cont.) Example: Convert 7.91×10‒7 to standard notation. Shift the decimal point 7 places to the left and fill in the blanks with 0s. .000000791 or 0.000000791
- 14. Classwork College Algebra 2e (OpenStax.org) ⚫ 1.2: 6-24 (even); 1.1: 28-56 (×4) ⚫ 1.2 Classwork Check (in Canvas) ⚫ Quiz 1.1