Scientific Notation

The standard form

M × 10ⁿ, where 1 ≤ M < 10 and n is an integer. Example: 4.7 × 10⁵ If M is 47 or 0.47, it is not in proper scientific notation yet.

Big numbers → positive exponent

Formula: move the decimal LEFT, exponent goes UP Example: 93,000,000 = 9.3 × 10⁷ The decimal moved 7 places left, so n = +7.

Small numbers → negative exponent

Formula: move the decimal RIGHT, exponent goes DOWN Example: 0.00042 = 4.2 × 10⁻⁴ The decimal moved 4 places right, so n = −4.

Multiplying

Formula: (a × 10ᵐ)(b × 10ⁿ) = (a·b) × 10ᵐ⁺ⁿ Example: (2.0 × 10³)(4.0 × 10⁵) = 8.0 × 10⁸ Multiply the coefficients, ADD the exponents.

Dividing

Formula: (a × 10ᵐ) ÷ (b × 10ⁿ) = (a÷b) × 10ᵐ⁻ⁿ Example: (8.0 × 10⁶) ÷ (2.0 × 10²) = 4.0 × 10⁴ Divide the coefficients, SUBTRACT the exponents.

Adding and subtracting

Rule: exponents must MATCH first. Example: 3.0 × 10⁴ + 2.0 × 10³ = 3.0 × 10⁴ + 0.20 × 10⁴ = 3.2 × 10⁴ Rewrite the smaller number to share the larger exponent, then add the coefficients.

Re-normalizing the answer

If the coefficient lands outside 1–10, fix it. Example: 12.5 × 10⁷ = 1.25 × 10⁸ and 0.42 × 10⁻³ = 4.2 × 10⁻⁴ Every place you move the decimal changes the exponent by one.

Significant figures

Only the coefficient carries sig figs. Example: 2.50 × 10⁴ has 3 sig figs; 2.5 × 10⁴ has 2. The power of ten is a placeholder, never a significant digit.

Powers and roots

Formula: (a × 10ⁿ)^p = a^p × 10ⁿ·ᵖ Example: (3.0 × 10⁴)² = 9.0 × 10⁸ Raise the coefficient to the power and MULTIPLY the exponent by it.

Numbers you should recognize

Avogadro's number: 6.022 × 10²³ particles per mole Speed of light: 3.00 × 10⁸ m/s Planck's constant: 6.626 × 10⁻³⁴ J·s Charge of an electron: 1.602 × 10⁻¹⁹ C

Calculator habit

Use the EE or EXP key, not ×10^. Example: 6.022 × 10²³ is typed 6.022 EE 23 Typing "6.022 × 10 ^ 23" without parentheses is the #1 source of wrong answers.

10⁰ = 1

Formula: any number × 10⁰ = itself Example: 7.4 × 10⁰ = 7.4 A zero exponent means the decimal never moved.