Exponent Arithmetic

U.S. electricity generation is about 4 x 10¹² kWh per year. A single household uses about 10⁴ kWh per year. How many households is that? If you can subtract exponents, the answer takes three seconds: 10¹² / 10⁴ = 10⁸, or about 100 million. Roughly correct, no calculator needed.

Two rules handle almost every energy-scale calculation. Multiplication: add the exponents. 10⁶ x 10³ = 10⁹. A megawatt (10⁶ watts) running for a thousand hours (10³) delivers 10⁹ watt-hours, or one gigawatt-hour. Division: subtract the exponents. 10¹⁵ / 10⁹ = 10⁶. A petajoule divided by a gigajoule gives a million units.

When the coefficients are not exactly 1, handle them separately. (3 x 10⁸) x (2 x 10⁴) = (3 x 2) x 10⁸⁺⁴ = 6 x 10¹². The coefficients multiply normally; the exponents add.

For negative exponents, the same rules apply. 10⁶ x 10⁻² = 10⁴. This comes up when converting units downward (from mega to kilo, for instance).

Worked Example

Global primary energy is about 5.8 x 10²⁰ joules per year. World population is about 8 x 10⁹ people.

• Divide coefficients. 5.8 / 8 = 0.725.
• Subtract exponents. 10²⁰ / 10⁹ = 10¹¹.
• Combine. 0.725 x 10¹¹ = 7.25 x 10¹⁰ joules per person per year.

Is that a big number? Convert to gigajoules (10⁹ J) for a more intuitive feel.

7.25 x 10¹⁰ / 10⁹ = 72.5 GJ per person per year. About 73 gigajoules. That is roughly the energy in 600 gallons of gasoline, consumed across all activities: heating, transport, electricity, food production, and industry.

A calculator gets the same answer. What exponent arithmetic buys is speed: the difference between checking a claim during a meeting and looking foolish after it ends.

---