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Approximations worth memorizing

$\sqrt2 \approx 1.414$ , $\sqrt3 \approx 1.732$
$2^{10} = 1024 \approx 1000$ (useful for estimating large powers of 2)
$\pi \approx 3.14159$

Comparing growth rates

For large inputs, exponential functions always eventually outgrow polynomial ones, which always outgrow logarithmic ones — a fast sanity check when comparing the size of two expressions without computing either exactly.

Bounding tools

The AM-GM inequality and (for AMC 12) Cauchy-Schwarz are the most common general-purpose bounding tools — both let you constrain the range of an expression without solving for its exact value, which is often all a problem actually requires.

When estimation pays off

Estimation is worth doing on essentially every problem as a large-number sanity check on your final answer. It’s more selectively useful mid-solution — reach for it on messy expressions where an exact computation would eat significant time, and consider testing extreme inputs (0, 1, −1, or a very large number) to get a feel for an expression’s behavior before committing to a full solve.

Applying it

Scenario: A problem asks you to estimate $2^{33}$ to determine how many digits it has.

Response: Use $2^{10} \approx 10^3$ : $2^{33} = 2^{30}\cdot2^3 \approx (10^3)^3 \cdot 8 = 8 \times 10^9$ , which has 10 digits. (The true value, $8{,}589{,}934{,}592$ , confirms this.)

Practice problems

1.Estimate $2^{20}$ using $2^{10}\approx1000$ .

2.Without computing exactly, which grows faster as n gets large: n^100 or 2^n?

3.Use $\sqrt2\approx1.414$ to estimate $\sqrt{8}$ .

4.A problem's final expression is a nested radical that's hard to simplify exactly, but the answer choices are far apart in value. What's the fastest way to proceed?

Estimation & Bounds

Approximations worth memorizing

Comparing growth rates

Bounding tools

When estimation pays off

Practice problems