Worked Examples
Four real problems, solved with real numbers — the same exponents, square roots, and scientific notation students meet in this module.
Figures current · August 2026
A = P(1 + r)ᵗt = time
Top high-yield savings accounts pay around 4% APY, while the national average savings rate is just 0.38%. Put $1,000 in at 4% and let the exponent do the work.
$1,216.65after 5 yearssimple interest: $1,200
→The exponent t compounds growth on top of growth — time, not the rate, does most of the lifting.
(1 + r)ᵗ = 2 → 72 ÷ rateshortcut
How long until money doubles? Solve for the exponent — or use the Rule of 72 shortcut and divide 72 by the interest rate for a close answer in your head.
18 yrsto double at 4%72 ÷ 4
→Small changes to the base of the exponent make an enormous difference over a lifetime.
σ = √variancerisk
An investment returns 4%, 6%, 8%, 10%, and 12% over five years. The average return is 8% and the variance is 8 — the square root turns that into something you can read.
±2.8 ptsstandard deviationmean 8%
→Standard deviation is the financial formula analysts use to measure how volatile, or risky, an investment's returns really are.
$39,800,000,000,000 = 3.98 × 10¹³
The U.S. national debt is about $39.8 trillion. Writing enormous financial figures as a base of ten raised to an exponent makes them far easier to compare.
11orders of magnitude between a $400 paycheck and U.S. GDP
→Every step right on this ruler is ten times bigger than the last.
Sources: FDIC national average savings rate; U.S. Treasury debt data. Figures reviewed August 2026.
