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Math Coach Amy
Amy Ferguson Moncure

7th Grade Math · Unit 2

Square Roots of
Perfect Squares

Find positive square roots, compare square roots, and find the side length of a square.

Jump to section:

Key Ideas Perfect Squares 0-100 Perfect Squares 121-400 Area and Side Length Closest Integers Comparing Add or Subtract Square Roots A Square and a Root Bigger Square Roots Square Root Fractions Quiz
What to Remember

Perfect Squares

0² = 0

1² = 1

2² = 4

3² = 9

4² = 16

5² = 25

6² = 36

7² = 49

8² = 64

9² = 81

10² = 100

11² = 121

12² = 144

13² = 169

14² = 196

15² = 225

16² = 256

17² = 289

18² = 324

19² = 361

20² = 400

Easy

Perfect Squares 0-100

Find the positive square root.

Example
Find the positive square root. square root of 16

If a positive number times itself is 16, that number is the positive square root.

4 × 4 = 16.

square root of 16 = 4

Answer: 4

1
Find the positive square root. square root of 64
Show Answer

If a positive number times itself is 64, that number is the positive square root.

8 × 8 = 64.

square root of 64 = 8

Answer: 8

2
Find the positive square root. square root of 9
Show Answer

If a positive number times itself is 9, that number is the positive square root.

3 × 3 = 9.

square root of 9 = 3

Answer: 3

3
Find the positive square root. square root of 100
Show Answer

If a positive number times itself is 100, that number is the positive square root.

10 × 10 = 100.

square root of 100 = 10

Answer: 10

4
Find the positive square root. square root of 1
Show Answer

If a positive number times itself is 1, that number is the positive square root.

1 × 1 = 1.

square root of 1 = 1

Answer: 1

5
Find the positive square root. square root of 49
Show Answer

If a positive number times itself is 49, that number is the positive square root.

7 × 7 = 49.

square root of 49 = 7

Answer: 7

6
Find the square root. square root of 0
Show Answer

0 × 0 = 0.

square root of 0 = 0

Answer: 0

7
Find the positive square root. square root of 36
Show Answer

If a positive number times itself is 36, that number is the positive square root.

6 × 6 = 36.

square root of 36 = 6

Answer: 6

8
Find the positive square root. square root of 81
Show Answer

If a positive number times itself is 81, that number is the positive square root.

9 × 9 = 81.

square root of 81 = 9

Answer: 9

9
Find the positive square root. square root of 4
Show Answer

If a positive number times itself is 4, that number is the positive square root.

2 × 2 = 4.

square root of 4 = 2

Answer: 2

10
Find the positive square root. square root of 25
Show Answer

If a positive number times itself is 25, that number is the positive square root.

5 × 5 = 25.

square root of 25 = 5

Answer: 5

Easy 2

Perfect Squares 121-400

Find the positive square root.

1
Find the positive square root. square root of 361
Show Answer

If a positive number times itself is 361, that number is the positive square root.

19 × 19 = 361.

square root of 361 = 19

Answer: 19

2
Find the positive square root. square root of 121
Show Answer

If a positive number times itself is 121, that number is the positive square root.

11 × 11 = 121.

square root of 121 = 11

Answer: 11

3
Find the positive square root. square root of 256
Show Answer

If a positive number times itself is 256, that number is the positive square root.

16 × 16 = 256.

square root of 256 = 16

Answer: 16

4
Find the positive square root. square root of 400
Show Answer

If a positive number times itself is 400, that number is the positive square root.

20 × 20 = 400.

square root of 400 = 20

Answer: 20

5
Find the positive square root. square root of 169
Show Answer

If a positive number times itself is 169, that number is the positive square root.

13 × 13 = 169.

square root of 169 = 13

Answer: 13

6
Find the positive square root. square root of 324
Show Answer

If a positive number times itself is 324, that number is the positive square root.

18 × 18 = 324.

square root of 324 = 18

Answer: 18

7
Find the positive square root. square root of 196
Show Answer

If a positive number times itself is 196, that number is the positive square root.

14 × 14 = 196.

square root of 196 = 14

Answer: 14

8
Find the positive square root. square root of 289
Show Answer

If a positive number times itself is 289, that number is the positive square root.

17 × 17 = 289.

square root of 289 = 17

Answer: 17

9
Find the positive square root. square root of 144
Show Answer

If a positive number times itself is 144, that number is the positive square root.

12 × 12 = 144.

square root of 144 = 12

Answer: 12

10
Find the positive square root. square root of 225
Show Answer

If a positive number times itself is 225, that number is the positive square root.

15 × 15 = 225.

square root of 225 = 15

Answer: 15

Medium

Area and Side Length

The side length is the positive square root of the area.

Example
A square poster has area 144 square centimeters. What is the side length in centimeters?
144square centimeters144square centimeters12 centimeters12 centimeters12 centimeters12 centimeters

The side length is the positive square root of the area.

Side length = square root of 144

12 × 12 = 144.

square root of 144 = 12

The side length is 12 centimeters.

Answer: 12 centimeters

1
A square garden has area 400 square meters. What is the side length in meters?
Show Answer
400square meters400square meters20 meters20 meters20 meters20 meters

The side length is the positive square root of the area.

Side length = square root of 400

20 × 20 = 400.

square root of 400 = 20

The side length is 20 meters.

Answer: 20 meters

2
A square tile has area 9 square inches. What is the side length in inches?
Show Answer
9square inches9square inches3 inches3 inches3 inches3 inches

The side length is the positive square root of the area.

Side length = square root of 9

3 × 3 = 9.

square root of 9 = 3

The side length is 3 inches.

Answer: 3 inches

3
A square patio has area 289 square feet. What is the side length in feet?
Show Answer
289square feet289square feet17 feet17 feet17 feet17 feet

The side length is the positive square root of the area.

Side length = square root of 289

17 × 17 = 289.

square root of 289 = 17

The side length is 17 feet.

Answer: 17 feet

4
A square window has area 64 square inches. What is the side length in inches?
Show Answer
64square inches64square inches8 inches8 inches8 inches8 inches

The side length is the positive square root of the area.

Side length = square root of 64

8 × 8 = 64.

square root of 64 = 8

The side length is 8 inches.

Answer: 8 inches

5
A square table has area 25 square feet. What is the side length in feet?
Show Answer
25square feet25square feet5 feet5 feet5 feet5 feet

The side length is the positive square root of the area.

Side length = square root of 25

5 × 5 = 25.

square root of 25 = 5

The side length is 5 feet.

Answer: 5 feet

Medium 2

Closest Integers

The number is not a perfect square.

Example
square root of 39 falls between what two integers?

6 × 6 = 36.

7 × 7 = 49.

39 is between 36 and 49.

square root of 39 is between 6 and 7.

Answer: 6 and 7

1
square root of 90 falls between what two integers?
Show Answer

9 × 9 = 81.

10 × 10 = 100.

90 is between 81 and 100.

square root of 90 is between 9 and 10.

Answer: 9 and 10

2
square root of 3 falls between what two integers?
Show Answer

1 × 1 = 1.

2 × 2 = 4.

3 is between 1 and 4.

square root of 3 is between 1 and 2.

Answer: 1 and 2

3
square root of 30 falls between what two integers?
Show Answer

5 × 5 = 25.

6 × 6 = 36.

30 is between 25 and 36.

square root of 30 is between 5 and 6.

Answer: 5 and 6

4
square root of 75 falls between what two integers?
Show Answer

8 × 8 = 64.

9 × 9 = 81.

75 is between 64 and 81.

square root of 75 is between 8 and 9.

Answer: 8 and 9

5
square root of 20 falls between what two integers?
Show Answer

4 × 4 = 16.

5 × 5 = 25.

20 is between 16 and 25.

square root of 20 is between 4 and 5.

Answer: 4 and 5

Medium 3

Comparing

Compare two square roots, or a square root and an integer.

Example
Compare. Write <, >, or =.
square root of 49    ?    square root of 16

The larger number has the larger positive square root.

49 > 16.

square root of 49 > square root of 16

Answer: square root of 49 > square root of 16

Example
Compare. Write <, >, or =.
square root of 55    ?    8

Square the integer. Compare that square with the number under the radical.

8 × 8 = 64.

55 < 64.

square root of 55 < 8

Answer: square root of 55 < 8

1
Compare. Write <, >, or =.
square root of 9    ?    square root of 49
Show Answer

The larger number has the larger positive square root.

9 < 49.

square root of 9 < square root of 49

Answer: square root of 9 < square root of 49

2
Compare. Write <, >, or =.
square root of 81    ?    7
Show Answer

Square the integer. Compare that square with the number under the radical.

7 × 7 = 49.

81 > 49.

square root of 81 > 7

Answer: square root of 81 > 7

3
Compare. Write <, >, or =.
square root of 121    ?    square root of 100
Show Answer

The larger number has the larger positive square root.

121 > 100.

square root of 121 > square root of 100

Answer: square root of 121 > square root of 100

4
Compare. Write <, >, or =.
square root of 16    ?    5
Show Answer

Square the integer. Compare that square with the number under the radical.

5 × 5 = 25.

16 < 25.

square root of 16 < 5

Answer: square root of 16 < 5

5
Compare. Write <, >, or =.
square root of 45    ?    square root of 20
Show Answer

The larger number has the larger positive square root.

45 > 20.

square root of 45 > square root of 20

Answer: square root of 45 > square root of 20

6
Compare. Write <, >, or =.
square root of 36    ?    6
Show Answer

Square the integer. Compare that square with the number under the radical.

6 × 6 = 36.

The number under the radical matches.

square root of 36 = 6

Answer: square root of 36 = 6

7
Compare. Write <, >, or =.
square root of 144    ?    square root of 169
Show Answer

The larger number has the larger positive square root.

144 < 169.

square root of 144 < square root of 169

Answer: square root of 144 < square root of 169

8
Compare. Write <, >, or =.
square root of 48    ?    7
Show Answer

Square the integer. Compare that square with the number under the radical.

7 × 7 = 49.

48 < 49.

square root of 48 < 7

Answer: square root of 48 < 7

9
Compare. Write <, >, or =.
square root of 225    ?    square root of 196
Show Answer

The larger number has the larger positive square root.

225 > 196.

square root of 225 > square root of 196

Answer: square root of 225 > square root of 196

10
Compare. Write <, >, or =.
square root of 100    ?    10
Show Answer

Square the integer. Compare that square with the number under the radical.

10 × 10 = 100.

The number under the radical matches.

square root of 100 = 10

Answer: square root of 100 = 10

Hard

Add or Subtract Square Roots

Find each positive square root. Then add or subtract.

Example
Find the value. square root of 100 + square root of 49

Find each positive square root. Then add or subtract.

square root of 100 = 10 because 10 × 10 = 100.

square root of 49 = 7 because 7 × 7 = 49.

10 + 7 = 17

Answer: 17

1
Find the value. square root of 400 − square root of 225
Show Answer

Find each positive square root. Then add or subtract.

square root of 400 = 20 because 20 × 20 = 400.

square root of 225 = 15 because 15 × 15 = 225.

20 − 15 = 5

Answer: 5

2
Find the value. square root of 324 + square root of 36
Show Answer

Find each positive square root. Then add or subtract.

square root of 324 = 18 because 18 × 18 = 324.

square root of 36 = 6 because 6 × 6 = 36.

18 + 6 = 24

Answer: 24

3
Find the value. square root of 196 + square root of 4
Show Answer

Find each positive square root. Then add or subtract.

square root of 196 = 14 because 14 × 14 = 196.

square root of 4 = 2 because 2 × 2 = 4.

14 + 2 = 16

Answer: 16

4
Find the value. square root of 289 − square root of 49
Show Answer

Find each positive square root. Then add or subtract.

square root of 289 = 17 because 17 × 17 = 289.

square root of 49 = 7 because 7 × 7 = 49.

17 − 7 = 10

Answer: 10

5
Find the value. square root of 361 − square root of 121
Show Answer

Find each positive square root. Then add or subtract.

square root of 361 = 19 because 19 × 19 = 361.

square root of 121 = 11 because 11 × 11 = 121.

19 − 11 = 8

Answer: 8

6
Find the value. square root of 256 + square root of 64
Show Answer

Find each positive square root. Then add or subtract.

square root of 256 = 16 because 16 × 16 = 256.

square root of 64 = 8 because 8 × 8 = 64.

16 + 8 = 24

Answer: 24

7
Find the value. square root of 225 − square root of 81
Show Answer

Find each positive square root. Then add or subtract.

square root of 225 = 15 because 15 × 15 = 225.

square root of 81 = 9 because 9 × 9 = 81.

15 − 9 = 6

Answer: 6

8
Find the value. square root of 169 + square root of 16
Show Answer

Find each positive square root. Then add or subtract.

square root of 169 = 13 because 13 × 13 = 169.

square root of 16 = 4 because 4 × 4 = 16.

13 + 4 = 17

Answer: 17

9
Find the value. square root of 144 + square root of 25
Show Answer

Find each positive square root. Then add or subtract.

square root of 144 = 12 because 12 × 12 = 144.

square root of 25 = 5 because 5 × 5 = 25.

12 + 5 = 17

Answer: 17

10
Find the value. square root of 400 − square root of 100
Show Answer

Find each positive square root. Then add or subtract.

square root of 400 = 20 because 20 × 20 = 400.

square root of 100 = 10 because 10 × 10 = 100.

20 − 10 = 10

Answer: 10

Extra

A Square and a Square Root

Square the first number. Then find the positive square root. Then add or subtract.

Example
Simplify. 5² + square root of 25

Square the first number. Then find the positive square root. Then add or subtract.

5² = 5 × 5 = 25

square root of 25 = 5

25 + 5 = 30

Answer: 30

1
Simplify. 4² + square root of 4
Show Answer

Square the first number. Then find the positive square root. Then add or subtract.

4² = 4 × 4 = 16

square root of 4 = 2

16 + 2 = 18

Answer: 18

2
Simplify. 16² + square root of 16
Show Answer

Square the first number. Then find the positive square root. Then add or subtract.

16² = 16 × 16 = 256

square root of 16 = 4

256 + 4 = 260

Answer: 260

3
Simplify. 36² + square root of 36
Show Answer

Square the first number. Then find the positive square root. Then add or subtract.

36² = 36 × 36 = 1,296

square root of 36 = 6

1,296 + 6 = 1,302

Answer: 1,302

4
Simplify. 8² + square root of 49
Show Answer

Square the first number. Then find the positive square root. Then add or subtract.

8² = 8 × 8 = 64

square root of 49 = 7

64 + 7 = 71

Answer: 71

5
Simplify. 11² + square root of 36
Show Answer

Square the first number. Then find the positive square root. Then add or subtract.

11² = 11 × 11 = 121

square root of 36 = 6

121 + 6 = 127

Answer: 127

6
Simplify. 13² − square root of 16
Show Answer

Square the first number. Then find the positive square root. Then add or subtract.

13² = 13 × 13 = 169

square root of 16 = 4

169 − 4 = 165

Answer: 165

7
Simplify. 6² + square root of 64
Show Answer

Square the first number. Then find the positive square root. Then add or subtract.

6² = 6 × 6 = 36

square root of 64 = 8

36 + 8 = 44

Answer: 44

8
Simplify. 15² − square root of 121
Show Answer

Square the first number. Then find the positive square root. Then add or subtract.

15² = 15 × 15 = 225

square root of 121 = 11

225 − 11 = 214

Answer: 214

Challenge 2

Bigger Square Roots

Each number is a perfect square and a multiple of 10. Find the positive square root.

1
Find the positive square root. square root of 6400
Show Answer

80 = 8 × 10.

8 × 8 = 64.

10 × 10 = 100.

64 × 100 = 6,400.

80 × 80 = 6,400.

square root of 6400 = 80

Answer: 80

2
Find the positive square root. square root of 4900
Show Answer

70 = 7 × 10.

7 × 7 = 49.

10 × 10 = 100.

49 × 100 = 4,900.

70 × 70 = 4,900.

square root of 4900 = 70

Answer: 70

3
Find the positive square root. square root of 14400
Show Answer

120 = 12 × 10.

12 × 12 = 144.

10 × 10 = 100.

144 × 100 = 14,400.

120 × 120 = 14,400.

square root of 14400 = 120

Answer: 120

4
Find the positive square root. square root of 8100
Show Answer

90 = 9 × 10.

9 × 9 = 81.

10 × 10 = 100.

81 × 100 = 8,100.

90 × 90 = 8,100.

square root of 8100 = 90

Answer: 90

5
Find the positive square root. square root of 12100
Show Answer

110 = 11 × 10.

11 × 11 = 121.

10 × 10 = 100.

121 × 100 = 12,100.

110 × 110 = 12,100.

square root of 12100 = 110

Answer: 110

Challenge 3

Square Root Fractions

The numerator and the denominator are both perfect squares. Find the positive square root.

1
Find the positive square root. square root of 19
Show Answer

Take the positive square root of the numerator and of the denominator.

square root of 1 = 1 because 1 × 1 = 1.

square root of 9 = 3 because 3 × 3 = 9.

square root of 19 = 13

Answer: 13

2
Find the positive square root. square root of 425
Show Answer

Take the positive square root of the numerator and of the denominator.

square root of 4 = 2 because 2 × 2 = 4.

square root of 25 = 5 because 5 × 5 = 25.

square root of 425 = 25

Answer: 25

3
Find the positive square root. square root of 949
Show Answer

Take the positive square root of the numerator and of the denominator.

square root of 9 = 3 because 3 × 3 = 9.

square root of 49 = 7 because 7 × 7 = 49.

square root of 949 = 37

Answer: 37

4
Find the positive square root. square root of 1681
Show Answer

Take the positive square root of the numerator and of the denominator.

square root of 16 = 4 because 4 × 4 = 16.

square root of 81 = 9 because 9 × 9 = 81.

square root of 1681 = 49

Answer: 49

5
Find the positive square root. square root of 2564
Show Answer

Take the positive square root of the numerator and of the denominator.

square root of 25 = 5 because 5 × 5 = 25.

square root of 64 = 8 because 8 × 8 = 64.

square root of 2564 = 58

Answer: 58

Quiz

Quiz

One of each type, from easiest to hardest. For more practice, open the answer and use the section it names.

1
Find the positive square root. square root of 49
Show Answer

If a positive number times itself is 49, that number is the positive square root.

7 × 7 = 49.

square root of 49 = 7

For more practice, go to Perfect Squares 0-100.

Answer: 7

2
Find the positive square root. square root of 324
Show Answer

If a positive number times itself is 324, that number is the positive square root.

18 × 18 = 324.

square root of 324 = 18

For more practice, go to Perfect Squares 121-400.

Answer: 18

3
A square deck has area 196 square feet. What is the side length in feet?
Show Answer

The side length is the positive square root of the area.

Side length = square root of 196

14 × 14 = 196.

square root of 196 = 14

The side length is 14 feet.

For more practice, go to Area and Side Length.

Answer: 14 feet

4
square root of 60 falls between what two integers?
Show Answer

7 × 7 = 49.

8 × 8 = 64.

60 is between 49 and 64.

square root of 60 is between 7 and 8.

For more practice, go to Closest Integers.

Answer: 7 and 8

5
Find the value. square root of 121 + square root of 64
Show Answer

Find each positive square root. Then add or subtract.

square root of 121 = 11 because 11 × 11 = 121.

square root of 64 = 8 because 8 × 8 = 64.

11 + 8 = 19

For more practice, go to Add or Subtract Square Roots.

Answer: 19

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