Squares, cubes and square roots
| n | n² | n³ | √n |
|---|---|---|---|
| 1 | 1 | 1 | 1 |
| 2 | 4 | 8 | 1.414 |
| 3 | 9 | 27 | 1.732 |
| 4 | 16 | 64 | 2 |
| 5 | 25 | 125 | 2.236 |
| 6 | 36 | 216 | 2.449 |
| 7 | 49 | 343 | 2.646 |
| 8 | 64 | 512 | 2.828 |
| 9 | 81 | 729 | 3 |
| 10 | 100 | 1,000 | 3.162 |
| 11 | 121 | 1,331 | 3.317 |
| 12 | 144 | 1,728 | 3.464 |
| 13 | 169 | 2,197 | 3.606 |
| 14 | 196 | 2,744 | 3.742 |
| 15 | 225 | 3,375 | 3.873 |
| 16 | 256 | 4,096 | 4 |
| 17 | 289 | 4,913 | 4.123 |
| 18 | 324 | 5,832 | 4.243 |
| 19 | 361 | 6,859 | 4.359 |
| 20 | 400 | 8,000 | 4.472 |
| 21 | 441 | 9,261 | 4.583 |
| 22 | 484 | 10,648 | 4.690 |
| 23 | 529 | 12,167 | 4.796 |
| 24 | 576 | 13,824 | 4.899 |
| 25 | 625 | 15,625 | 5 |
| 26 | 676 | 17,576 | 5.099 |
| 27 | 729 | 19,683 | 5.196 |
| 28 | 784 | 21,952 | 5.292 |
| 29 | 841 | 24,389 | 5.385 |
| 30 | 900 | 27,000 | 5.477 |
| 31 | 961 | 29,791 | 5.568 |
| 32 | 1024 | 32,768 | 5.657 |
| 33 | 1089 | 35,937 | 5.745 |
| 34 | 1156 | 39,304 | 5.831 |
| 35 | 1225 | 42,875 | 5.916 |
| 36 | 1296 | 46,656 | 6 |
| 37 | 1369 | 50,653 | 6.083 |
| 38 | 1444 | 54,872 | 6.164 |
| 39 | 1521 | 59,319 | 6.245 |
| 40 | 1600 | 64,000 | 6.325 |
| 41 | 1681 | 68,921 | 6.403 |
| 42 | 1764 | 74,088 | 6.481 |
| 43 | 1849 | 79,507 | 6.557 |
| 44 | 1936 | 85,184 | 6.633 |
| 45 | 2025 | 91,125 | 6.708 |
| 46 | 2116 | 97,336 | 6.782 |
| 47 | 2209 | 103,823 | 6.856 |
| 48 | 2304 | 110,592 | 6.928 |
| 49 | 2401 | 117,649 | 7 |
| 50 | 2500 | 125,000 | 7.071 |
Squares, cubes and square rootsstringmash.com/square-numbers-chart
Reading the chart
Each row has a number, its square (the number times itself), its cube (times itself twice) and its square root to three decimal places. Bold is the square.
To take a root, read a row backwards. 576 is in the squares column next to 24, so √576 = 24. For a number that isn't there, type it in the box above. It simplifies the root, so √72 comes out as 6√2.
Which squares to learn
Most courses expect the squares up to 12 × 12 = 144, the corner of the multiplication chart. Learn them to 15 or 20 and square roots, Pythagoras questions and estimating with big numbers all get noticeably quicker, because you spot the square straight away instead of working it out.
Tick blank to hide the squares column and print a practice sheet. The roots stay as clues.
Patterns worth spotting
The gaps between squares are the odd numbers. 1, 4, 9, 16 and 25 go up by 3, 5, 7 and 9, so adding 2n + 1 to n² always gives (n + 1)². Neat.
A square never ends in 2, 3, 7 or 8. If a number does, it isn't a perfect square and its root is irrational.
Questions
Is 0 a square number?
Yes, 0 × 0 = 0. Charts usually start at 1, but the calculator above accepts 0.
What is a perfect square?
Another name for a square number: a whole number times itself. 49 is a perfect square, 50 is not.
What does simplest radical form mean?
Taking the biggest square factor out of the root. 72 = 36 × 2, and √36 = 6, so √72 = 6√2.
Sources
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