Operating on Numbers in Scientific Notation

Scientific notation is a form of displaying very large/small numbers in a compact way. Numbers in scientific notation is written in the form a×10n. Here's how to do basic operations on numbers in scientific notation.

Addition
(Ex. Add 3×105 to 4.1×106.) 3×105 + 4.1×106 = 4.4×106, or 4,400,000 in ordinary notation.
 * 1) Adjust the numbers to have the same power of 10. 3×105 = 0.3×106.
 * 2) Add 0.3 to 4.1. 4.1 + 0.3 = 4.4.
 * 3) If the sum is 10 or greater, shift it until it is between 1 and 10.
 * 4) Add the number of places shifted to the exponent.

Subtraction
(Ex. Subtract 7.3×103 from 3.2×104.) 3.2×104 – 7.3×103 = 2.47×104, or 24,700 in ordinary notation.
 * 1) Adjust the numbers to have the same power of 10. 7.3×103 = 0.73×104.
 * 2) Subtract 0.73 from 3.2. 3.2 – 0.73 = 2.47.
 * 3) If the difference is less than 1, shift it until it is between 1 and 10. Subtract the number of places shifted from the exponent.

Multiplication
(Ex. Multiply 7.2×1011 by 2.4×108.) 7.2×1011 × 2.4×108 = 1.728×1020, or 172,800,000,000,000,000,000 (172.8 quintillion) in ordinary notation.
 * 1) Multiply 7.2 by 2.4. 7.2×2.4 = 17.28.
 * 2) If the result is less than 1 or at least 10, shift it until it is between 1 and 10.
 * 3) Add 11 to 8. 11 + 8 = 19. If you shifted any decimal places, adjust the exponent; if you shifted right, subtract the amount of places shifted from the exponent; if you shifted left, add the amount of places shifted to the exponent.

Division
(Ex. Divide 1.05×1012 by 1.5×107.) 1.05×1012 ÷ 1.5×107 = 7×104, or 70,000 in ordinary notation.
 * 1) Divide 1.05 by 1.5. 1.05 ÷ 1.5 = 0.7.
 * 2) If the result is less than 1 or at least 10, shift it until it is between 1 and 10.
 * 3) Subtract 7 from 12. 12 - 7 = 5. If you shifted any decimal places, adjust the exponent; if you shifted right, subtract the amount of places shifted from the exponent; if you shifted left, add the amount of places shifted to the exponent.

Exponentiation
(Ex. Raise 1.1×1012 to the 4th power.) (1.1×1012)4 = 1.4641×1048.
 * 1) Raise 1.1 to the 4th power. 1.14 = 1.4641.
 * 2) If the result is less than 1 or at least 10, shift it until it is between 1 and 10.
 * 3) Multiply 12 by 4. 12×4 = 48. If you shifted any decimal places, adjust the exponent; if you shifted right, subtract the amount of places shifted from the exponent; if you shifted left, add the amount of places shifted to the exponent.

Square Roots
(Ex. Find the square root of 4×108.) √(4×108) = 2×104, or 20,000 in ordinary notation.
 * 1) Take the square root of the first part. √4 = 2.
 * 2) Divide the exponent by 2. 8 ÷ 2 = 4.