Reference Table: Square Roots 1 to 20
Precise square roots are foundational for algebra, geometry, and number sense. The table below gives exact radical forms and practical decimals for the integers 1 through 20, alongside key patterns that support lasting recall.
Why Exact Forms Matter
Exact radical forms preserve precision, unlike truncated decimals. In algebra and proofs, expressing √8 as 2√2 shows structure and supports cancellation, while 2.828 is an approximation. For perfect squares—1, 4, 9, 16—the root is an integer. For non-squares, the radical stays as a concise, repeatable representation. This exact-first habit reduces small rounding errors in multi-step work and keeps results comparable across problems.
How to Simplify Square Roots Quickly
To simplify √n, factor n into primes, then pair identical factors. Each pair comes out of the radical as one copy. For example, √20 = √(2·2·5) = 2√5. On the 1–20 list, only the perfect squares simplify to integers; numbers like 8 and 18 reduce to multiples of smaller radicals. Memorizing key radicals (√2, √3, √5) helps you recognize simplifications fast and avoids unnecessary decimals early in a calculation.
Practical Examples
Use exact roots whenever a problem expects an algebraic answer. With a right triangle having legs of length 1 and 1, the hypotenuse is √2, not 1.414. If you double both legs, the hypotenuse becomes 2√2, matching √8. These patterns show why radicals are more informative than rounded decimals in geometry and measurement. When decimals are needed, convert the exact form at the last step to retain accuracy.
Patterns and Memory Aids
Notice how square roots repeat structure: √8 and √18 both involve √2 because 8 = 4·2 and 18 = 9·2. Similarly, √12 and √20 share √3 and √5 after factoring. Grouping numbers by their simplified radical family—Integers, √2 family, √3 family, √5 family, and others—makes the 1–20 list easier to remember. Practice by writing the exact forms from memory, then checking against the table until the patterns feel intuitive.
Beyond 1 to 20
Once the 1–20 list is familiar, extend the same method to larger numbers: factor, pair, and pull out squares. The habits used here—exact forms, simplification, and recognizing repeated radicals—scale to more advanced algebra, calculus, and statistics. By treating square roots as expressions rather than only decimals, you build a durable foundation for higher math and standardized tests where precision and speed matter together.
Use the table as a quick reference and learning tool; combine exact forms with brief decimal checks to strengthen number sense and long-term recall.