Scientific Notation and Rates

• Write a number in scientific notation (a × 10ⁿ with 1 ≤ a < 10) and convert it back to standard form
• Multiply and divide numbers in scientific notation by handling the coefficients and the powers of 10 separately, then re-normalizing
• Compare numbers in scientific notation by looking at the power of 10 first, then the coefficient
• Find a unit rate (the amount per one unit) and use it to compare two options, such as price per ounce or miles per gallon
• Scale a unit rate up or down to answer a 'how many in t hours' or 'how long for n items' question

Scientific notation writes a number as a × 10ⁿ, where 1 ≤ a < 10 and n is an integer. Moving the decimal point left makes the exponent positive (large numbers): 4,500,000 = 4.5 × 10⁶. Moving it right makes the exponent negative (small numbers): 0.00052 = 5.2 × 10⁻⁴. To multiply, multiply the coefficients and add the exponents: (3 × 10⁴)(2 × 10⁵) = 6 × 10⁹. To divide, divide the coefficients and subtract the exponents: (8 × 10⁶) / (2 × 10²) = 4 × 10⁴. If the coefficient leaves the 1-to-10 range, re-normalize: (4 × 10³)(5 × 10⁴) = 20 × 10⁷ = 2 × 10⁸. To add or subtract, first rewrite both numbers with the same power of 10. To compare, the power of 10 decides first: 3 × 10⁸ is larger than 9 × 10⁷, and any positive number with a negative exponent is less than 1. A unit rate is a rate with a denominator of 1: divide the first quantity by the second. 336 miles on 12 gallons is 28 miles per gallon; $4.20 for 12 ounces is $0.35 per ounce. Unit rates let you compare options directly: $4.20 for 12 ounces ($0.35 per ounce) costs more per ounce than $5.60 for 20 ounces ($0.28 per ounce), so the larger size is the better buy. Once you have a unit rate, multiply to scale up (28 miles per gallon × 9 gallons = 252 miles) or divide to find how much is needed (420 miles ÷ 28 miles per gallon = 15 gallons). To convert a rate, multiply by unit fractions so the unwanted units cancel: 15 miles per hour × 5,280 feet per mile ÷ 3,600 seconds per hour = 22 feet per second, and convert before you compare two rates given in different units. Keep the units written in every step; a rate written upside down (gallons per mile instead of miles per gallon) is the most common error, and the ACT puts that answer among the choices.

• Count decimal places moved, and check the direction: a number smaller than 1 must have a negative exponent.
• After multiplying or dividing, check that the coefficient is still between 1 and 10. If not, shift the decimal and adjust the exponent.
• Say the unit rate aloud with its units ('dollars per ounce'). If the words are backwards, so is the division.
• A calculator can verify a scientific notation product, but the exponent rules are faster for clean numbers.