Squeeze the root
Trap a square root between two perfect squares.
A game of roots
Aribam says 250. Bijou must say its square root, but 250 is not a perfect square.
Roughly how big is the square root of 250?
What this lesson covers
The idea
Place the number between two known perfect squares to bound its square root, narrow the interval by squaring a number in between, and use the units digit to pick candidates; for a non-square, the nearer square gives an approximate root.
A game of roots
Aribam says 250. Bijou must say its square root, but 250 is not a perfect square.
Roughly how big is the square root of 250?
- About 12
- About 16
- About 25
Squeeze it
Slide the guess. The toy squares it and says too small or too big. First find √1936 exactly, then squeeze √250.
Bracket, narrow, pick
Place the number between two known perfect squares to bound its root. Narrow the interval by squaring a number in between, and use the units digit to pick candidates. For a non-square, the nearer square gives an approximate root.
√250: 15² = 225 and 16² = 256, so 15 < √250 < 16. 256 is much nearer to 250 than 225, so √250 is about 16.
- √1936 | Reason
- 40² = 1600 < 1936 < 2500 = 50² | 40 < √1936 < 50
- 1936 ends in 6 | root ends in 4 or 6
- 45² = 2025 > 1936 | 40 < √1936 < 45, so 44
Notes
To estimate a square root, put the number between two perfect squares, narrow the interval, and use the units digit. For a non-square, the nearer square gives the approximate root.
Check yourself
Between which two whole numbers is √500?
1521 lies between 30² = 900 and 40² = 1600, and ends in 1, so its root ends in 1 or 9. What is √1521?
Answer: 39
39 × 39 = 1521, so √1521 = 39.
Estimate √200 as a whole number. (14² = 196 and 15² = 225.)
Answer: 14
200 − 196 = 4, but 225 − 200 = 25. 196 is nearer, so √200 is about 14.
A square cloth has an area of 125 sq cm. What is the side of the largest square handkerchief with a whole-number side that can be cut from it?
Answer: 11 cm
11² = 121 fits in 125, but 12² = 144 does not. The largest side is 11 cm.
- 21 and 22. 22² = 484 is still less than 500, so the root is bigger than 22.
- 22 and 23 — correct. Yes! 22² = 484 < 500 < 529 = 23².
- 24 and 25. 24² = 576 is already bigger than 500.
- 20 and 21. 21² = 441 is less than 500, so the root is bigger than 21.