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GMAT Strategy
Roots, Squares and Cubes
Use common roots and perfect powers as numerical anchors for comparison, simplification, and estimation.
Purpose
Perfect powers and common roots are speed tools.
Recognizing them lets you simplify expressions and bracket unfamiliar values without a calculator.
Common root approximations
- sqrt(2)
- about 1.414
- sqrt(3)
- about 1.732
- sqrt(5)
- about 2.236
- sqrt(10)
- about 3.162
Estimation
Bracket a root between nearby perfect squares.
Because 7^2 = 49 and 8^2 = 64, sqrt(55) must lie between 7 and 8.
Useful squares 1-10
- 1-5
- 1, 4, 9, 16, 25
- 6-10
- 36, 49, 64, 81, 100
Useful squares 11-20
- 11-15
- 121, 144, 169, 196, 225
- 16-20
- 256, 289, 324, 361, 400
Useful cubes 1-10
- 1-5
- 1, 8, 27, 64, 125
- 6-10
- 216, 343, 512, 729, 1000
Reference
Higher cubes are mainly recognition anchors.
11^3 = 1331, 12^3 = 1728, 15^3 = 3375, and 20^3 = 8000 are useful landmarks when a question presents perfect cubes.
Mini application
Try this: which is larger, sqrt(5) or 2.2?
Use the root approximation or square both positive values.
Solution
sqrt(5) is larger.
sqrt(5) is about 2.236, and 2.2^2 = 4.84 < 5.
Remember this
Use perfect powers as landmarks.
They turn many root and exponent questions into recognition instead of computation.