Start with a square you know
29² = (30 − 1)²
30² − 30 − 30 + 1 = 841
To find 29², start with a 30 by 30 square. Its area is 900.
Remove a strip of width 1 from each of two adjacent sides. Each strip has area 30. Their 1 by 1 corner was counted twice, so add 1 back.
That leaves 29 by 29: 900 − 60 + 1 = 841. The final correction is added, even though 29 is below 30.
Use the last digit
35²
3 × 4 = 12
12 × 100 + 25 = 1225
For a positive whole number ending in 5, multiply the number before the final 5 by the next whole number. Then write 25 after the result.
For 35², multiply 3 by 4 to get 12. Writing 25 after 12 gives 1225. This shortcut applies to numbers ending in 5.
Bound a root, then check it
49 < 50 < 64
7 < √50 < 8
7.1² = 50.41
A square root undoes a square. Since 49 < 50 < 64, the square root of 50 lies between 7 and 8.
50 is close to 49, so try 7.1. Squaring gives 50.41, a little above 50. The estimate is a little too large; it is not an exact answer.
Complete the correction
28² = 900 − 120 + □
28 is 2 below 30. The two strips remove 2 × 30 × 2 = 120.
What belongs in the box?
- 4
- 2
- −4
Show the answer
4
The corner is 2 by 2, so add 4 back. Then 900 − 120 + 4 = 784.
Repair a mistaken sign
29² = 900 − 60 − 1 = 839
Someone has used the nearby-square method, but one sign is wrong.
Which change repairs the calculation?
- Change −60 to +60.
- Change −1 to +1.
- Change 900 to 841.
Show the answer
Change −1 to +1.
Add the overlapping corner back: 900 − 60 + 1 = 841.
Locate the root first
√70
Find the whole numbers on either side of √70.
Which interval contains √70?
- 7 < √70 < 8
- 9 < √70 < 10
- 8 < √70 < 9
Show the answer
8 < √70 < 9
8² = 64 and 9² = 81. Since 64 < 70 < 81, we have 8 < √70 < 9.
Check your estimate
8.4² = 70.56
You try 8.4 as an estimate of √70. Square it to see which side of the true root it lies on.
What does this check tell you about 8.4?
- It is a little too large.
- It is exact.
- It is a little too small.
Show the answer
It is a little too large.
70.56 is above 70. For positive numbers, a larger square means a larger number, so 8.4 is a little above √70.
Choose your own method
49²
Work this out without a hint. Choose a method you can explain.
What is 49²?
- 2501
- 2401
- 2399
Show the answer
2401
One method is (50 − 1)² = 2500 − 100 + 1 = 2401.
Try another square
75²
Choose an efficient method and calculate the answer.
What is 75²?
- 5650
- 5525
- 5625
Show the answer
5625
7 × 8 = 56. Write 25 after 56: 5625.
Use a root in a real problem
196 m²
A square garden has an area of 196 m². A fence will go around all four sides.
What are the side length and the total fence length?
- Side 14 m; fence 56 m.
- Side 14 m; fence 28 m.
- Side 49 m; fence 196 m.
Show the answer
Side 14 m; fence 56 m.
14 × 14 = 196, so each side is 14 m. Four sides need 4 × 14 = 56 m of fence.
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Practice Squares and roots