Scientific Notation and Integer Exponents
Students use exponent properties and scientific notation to represent, compare, and calculate with very large and very small quantities.
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Powers of Ten
A power of ten is written as 10 raised to an integer exponent. A positive exponent tells how many factors of 10 are multiplied: 10^4 = 10 × 10 × 10 × 10 = 10,000. The exponent also shows how many places the decimal point moves to the right from 1. A negative exponent represents a reciprocal. For example, 10^-3 = 1/10^3 = 1/1,000 = 0.001, so the decimal point moves three places to the left from 1. Also, 10^0 = 1. Each increase of 1 in the exponent makes the value ten times as large. Each decrease of 1 makes it one-tenth as large. This pattern helps represent quantities ranging from enormous astronomical distances to extremely small measurements.
Integer Exponent Rules
Exponent rules make expressions with the same nonzero base easier to simplify. When multiplying powers with the same base, add the exponents: a^m × a^n = a^(m+n). For example, 2^3 × 2^4 = 2^7 = 128. When dividing powers with the same base, subtract the exponents: a^m ÷ a^n = a^(m-n). Thus, 5^6 ÷ 5^2 = 5^4 = 625. To raise a power to another power, multiply the exponents: (a^m)^n = a^(mn). For instance, (3^2)^4 = 3^8. A zero exponent gives a^0 = 1, and a negative exponent gives a^-n = 1/a^n. These rules apply when the base is nonzero wherever division or a negative exponent is involved.
Writing Scientific Notation
Scientific notation expresses a number as a × 10^n, where 1 ≤ |a| < 10 and n is an integer. The number a is called the coefficient. To write 48,300,000 in scientific notation, move the decimal point seven places left to form 4.83. Because the original number is greater than 10, use a positive exponent: 48,300,000 = 4.83 × 10^7. For a small number, move the decimal point right and use a negative exponent. In 0.000072, the decimal moves five places right to form 7.2, so 0.000072 = 7.2 × 10^-5. Check that the coefficient’s absolute value is at least 1 but less than 10. The exponent records the number and direction of decimal-place moves.
Operations in Scientific Notation
To multiply numbers in scientific notation, multiply the coefficients and add the exponents. For example, (3 × 10^4)(2 × 10^5) = 6 × 10^9. To divide, divide the coefficients and subtract the exponents: (8 × 10^7) ÷ (4 × 10^2) = 2 × 10^5. For addition or subtraction, first rewrite the numbers with the same power of ten. For example, 4.2 × 10^6 + 3 × 10^5 becomes 4.2 × 10^6 + 0.3 × 10^6 = 4.5 × 10^6. After any operation, make sure the coefficient is at least 1 and less than 10 in absolute value. If a product is 18 × 10^6, rename it as 1.8 × 10^7.
Interpreting Calculator Results
Calculators often display scientific notation using E instead of writing “× 10^.” For example, 6.02E23 means 6.02 × 10^23, not 6.02 multiplied by 23. Similarly, 4.7E-8 means 4.7 × 10^-8. To interpret a result, read the number before E as the coefficient and the number after E as the exponent of 10. Suppose a calculator shows 1.25E6 after a computation. This equals 1.25 × 10^6, or 1,250,000 in decimal notation. Check whether the sign and size are reasonable. A negative exponent usually represents an absolute value between 0 and 1, while a large positive exponent represents a large absolute value. Some calculators use EXP or EE instead of E, but the meaning is the same.