SAT Scientific Notation: Operations With Very Large and Very Small Numbers

Scientific notation expresses numbers in the form a×10^n, where 1≤a<10 and n is an integer. A positive exponent shifts the decimal right: 3.5×10^4=35,000. A negative exponent shifts the decimal left: 3.5×10^(-4)=0.00035. Converting to scientific notation requires moving the decimal until the coefficient falls between 1 and 10 and counting the places moved: moving left increases the exponent, moving right decreases it.

Two conversion examples: 0.000072=7.2×10^(-5) (decimal moved 5 places right, giving negative exponent). 9,400,000=9.4×10^6 (decimal moved 6 places left, giving positive exponent). A quick sanity check is to verify that very large numbers have positive exponents and very small numbers have negative exponents. If your converted number has the wrong sign on the exponent relative to whether the original was large or small, you have moved the decimal in the wrong direction.

To multiply in scientific notation, multiply the coefficients and add the exponents: (3×10^4)×(2×10^3)=6×10^7. To divide, divide the coefficients and subtract the exponents: (8×10^6)/(2×10^2)=4×10^4. After computing, always check whether the coefficient is still between 1 and 10. If it falls outside that range, adjust by multiplying or dividing by a power of 10 and updating the exponent accordingly.

Example adjustment: (5×10^3)×(4×10^2)=20×10^5. Since 20 is not between 1 and 10, rewrite as 2×10^1×10^5=2×10^6. Practice prompt: compute (6×10^(-3))/(2×10^2). Coefficients: 6/2=3. Exponents: -3-2=-5. Result: 3×10^(-5). After every multiplication or division in scientific notation, verify the coefficient is between 1 and 10 and adjust the exponent if it is not, since failing to adjust is the most common scientific notation arithmetic error on the SAT.

Adding or subtracting in scientific notation requires both terms to have the same exponent. To add 3×10^5 and 4×10^4, rewrite the second term: 3×10^5+0.4×10^5=3.4×10^5. Alternatively, convert both to standard form, add, and convert back. Comparing values is easier using exponents alone: 6×10^8 is larger than 9×10^7 because 10^8>10^7 regardless of the coefficients.

Three micro-examples: (1) 5×10^6+3×10^6=8×10^6 (same exponents, add coefficients directly). (2) 2×10^5+4×10^4=2×10^5+0.4×10^5=2.4×10^5 (rewrite smaller-exponent term first). (3) Compare 4.5×10^9 and 8×10^8: 10^9>10^8 so the first value is larger regardless of coefficients. When comparing two scientific notation values, always compare exponents first and only compare coefficients if the exponents are equal.

Data interpretation questions present tables or graphs with measurements in scientific notation and ask you to compare values, compute ratios, or describe relationships. Develop a sense of scale rather than always converting to full decimal form: 10^9 is roughly "billions," 10^6 is "millions," and 10^(-3) is "thousandths." This scale sense lets you compare values and eliminate unreasonable answer choices without performing full conversions.

Build fluency with a 5-day drill: each day, convert five values from standard to scientific notation and five from scientific to standard, then compare three pairs of scientific notation values using exponents alone. On day 5, work five data-interpretation problems using scientific notation values and time yourself. Students who can compare magnitudes by exponent alone without converting to standard form answer scientific notation data questions in under 20 seconds, making them easy points rather than time-consuming calculations on the SAT.