Scientific notation

Scientific notation expresses any number in the form a × 10ⁿ, where a (the coefficient) satisfies 1 ≤ a < 10 and n is an integer exponent. Large numbers get positive exponents — 5,600,000 becomes 5.6 × 10⁶ — while small decimals get negative exponents — 0.00042 becomes 4.2 × 10⁻⁴. The exponent counts how many places the decimal point moves to return the number to standard form.

To convert into scientific notation, move the decimal point until exactly one nonzero digit sits to its left, then count the moves: rightward moves make the exponent negative, leftward moves make it positive. To convert back, reverse the process. A number like 42 × 10³ is not proper scientific notation because 42 is not between 1 and 10 — rewrite it as 4.2 × 10⁴.

Arithmetic is where the notation earns its keep. To multiply, multiply the coefficients and add the exponents: (2 × 10³)(3 × 10⁵) = 6 × 10⁸. To divide, divide the coefficients and subtract the exponents. To add or subtract, first rewrite the numbers with the same exponent. These rules follow directly from the laws of exponents, and they let you handle quantities like Avogadro's number or the mass of an electron without writing strings of zeros.

Scientific notation shows up across standardized and professional exams. The ACT tests conversions and computation in its pre-algebra material, Praxis Core Math includes it in working with numbers, and the FE Mechanical exam assumes fluency with powers of 10 alongside logarithms in its mathematics coverage.