Representing Squares and Square Roots-Gr 8-Solved Examples

Solved Examples and Worksheet for Representing Squares and Square Roots

Q1Find two integers that satisfy the equation

$x$ ² = 361.

A. + 19 and - 19
B. 19 and 20
C. +17 and -17
D. None of the above

Solutions

Step: 1

$x$ ² = 361

[Original equation.]

Step: 2

$x$ =

$361$

[Apply square root on both sides.]

Step: 3

= ±

$(19×19)$

[If

$y$ ² =

$z$ , then

$y$ = ±

$z$ .]

Step: 4

= ± 19

Step: 5

So, + 19 and - 19 are the two integers that satisfy the equation.

Correct Answer is : + 19 and - 19

Q2The area of a square field is 2116 ft². Find the cost of fencing the field at $4 per ft.
A. $761
B. $786
C. $736
D. None of the above

Solutions

Step: 1

Area of a square field =

$side$ ²

Step: 2

side =

$Area$

Step: 3

$2116$

[Substitute.]

Step: 4

$46×46$ = 46

[Write 2116 as 46 × 46 and simplify.]

Step: 5

The perimeter of the square field = 4 ×

$side of the square field$

Step: 6

= 4 × 46 = 184

[Substitute and multiply.]

Step: 7

Cost of fencing the field =

$Perimeter of the field$ ×

$Cost of fencing per ft$

Step: 8

= 184 × 4 = 736

[Substitute and multiply.]

Step: 9

So, the cost of fencing the field is $736.

Correct Answer is : $736

Q3What is the area of the figure, if Area = Length × Width?

A. 156 square units
B. 192 square units
C. 133 square units
D. 145 square units

Solutions

Step: 1

Area = length x width

[Formula.]

Step: 2

= 8√3 x 8√3

[Substitute the value of length and width.]

Step: 3

= 64 x 3

[Simplify.]

Step: 4

= 192

[Multiply.]

Step: 5

The area of the figure is 192 square units.

Correct Answer is : 192 square units

Q4Find the side of a square, if its area is 12 square units.
A. 6√3
B. 6/√3
C. 2/√3
D. 2√3

Solutions

Step: 1

The area of the square =

$side$ ²

[Formula.]

Step: 2

The side of the square = √

$area of the square$

Step: 3

= √12

[Substitute the area of the square as 12.]

Step: 4

= √

$2 x 2 x 3$

[Write 12 as a product of prime factors.]

Step: 5

= 2√3

[Find the square root.]

Step: 6

The side of the square with an area of 12 square units is 2√3 units.

Correct Answer is : 2√3

Q5Which two integers satisfy the equation -

$y$ ² = - 1369?
A. 37 and 74
B. 74 and -74
C. 37 and -37
D. None of the above

Solutions

Step: 1

$y$ ² = -1369

[Original equation.]

Step: 2

$y$ ² = 1369

[Divide each side by -1.]

Step: 3

$y$ = √1369

[Apply square roots to each side.]

Step: 4

= ± 37

[Find the square root.]

Step: 5

The two integers that satisfy the equation -

$y$ ² = -1369 are 37 and -37.

Correct Answer is : 37 and -37

Q6Find the side of a square floor, if its area is 196 square meters.

A. 12 m
B. 14 m
C. 13 m
D. 15 m

Solutions

Step: 1

Let

$n$ be the side of the square floor.

Step: 2

The area of a square = side × side.

Step: 3

196 =

$n$ ×

$n$

[Substitute the values.]

Step: 4

$n$ ² = 196

[Multiply.]

Step: 5

√

$n$ ² = √196

[Square root each side.]

Step: 6

$n$ = 14

[Find the positive square root, since the side-length cannot be negative.]

Step: 7

The side-length of the square floor is 14 meters.

Correct Answer is : 14 m

Q7A school is in the shape of a square, and its area is 22500 ft². Find its perimeter.
A. 600 ft
B. 590 ft
C. 620 ft
D. 610 ft

Solutions

Step: 1

The area of the school is in the form of square.

Step: 2

The area of the school = 22500 ft²

Step: 3

The side of the school =

$22500$

[The side of square =

$area of the square$ .]

Step: 4

$150×150$

[Write 22500 as perfect square.]

Step: 5

= 150

[Simplify.]

Step: 6

The perimeter of the school = 4 × 150

[The perimeter of the square = 4 × side of a square.]

Step: 7

= 600 ft.

[Simplify.]

Step: 8

The perimeter of the school is 600 ft.

Correct Answer is : 600 ft

Q8The units place of a 3 digit number is the square of the first odd prime number and the digit in the 10's place is the square root of a perfect square between 34 and 43. What is the 3 digit number if it is a perfect square?
A. 169
B. 256
C. 196
D. 225
E. 144

Solutions

Step: 1

Units place digit is a square of a first odd prime number = 3² = 9.

Step: 2

10's place digit is a square root of a perfect square between 34 and 43 =

$36$ = 6.

Step: 3

The three digit number is a perfect square.

Step: 4

So, the 3 digit number is 169.

Correct Answer is : 169

Q9Find the side of a square floor, if its area is 256 square meters.

A. 16 m
B. 14 m
C. 17 m
D. 15 m

Solutions

Step: 1

Let

$n$ be the side of the square floor.

Step: 2

The area of a square = side × side.

Step: 3

256 =

$n$ ×

$n$

[Substitute the values.]

Step: 4

$n$ ² = 256

[Multiply.]

Step: 5

$n2$ =

$256$

[Square root each side.]

Step: 6

$n$ = 16

[Find the positive square root, since the side-length cannot be negative.]

Step: 7

The side-length of the square floor is 16 meters.

Correct Answer is : 16 m

Q10The units place of a 3 digit number is the square of the first odd prime number and the digit in the 10's place is the square root of a perfect square between 35 and 43. What is the 3 digit number if it is a perfect square?

A. 169
B. 225
C. 196
D. 144
E. 256

Solutions

Step: 1

Units place digit is a square of a first odd prime number = 3² = 9.

Step: 2

10's place digit is a square root of a perfect square between 35 and 43 =

$36$ = 6.

Step: 3

The three digit number is a perfect square.

Step: 4

So, the 3 digit number is 169.

Correct Answer is : 169

Q11Laura has a square garden of length 9 m on each side. Now she is forming a square garden whose area is

$15$ of the area of the old one. What will be the side length of the new garden?
A. 9

$m$
B. 4.02

$m$
C. 5.16

$m$
D. 7

$m$

Solutions

Step: 1

The length of Laura's old garden = 9

$m$

Step: 2

Area of garden = 9 × 9 = 81

[area of a square = side²]

Step: 3

Area of Laura's new garden =

$15$ of the area of the old garden

Step: 4

$15$ × 81 = 16.2

Step: 5

Side length of the new garden =

$area$ =

$16.2$ = 4.02

[side =

$area$ ]

Correct Answer is : 4.02

$m$

Q12Identify the square root that the point represents on the number line.

$8$
B.

$98$
C.

$146$
D.

$62$

Solutions

Step: 1

The perfect square nearer to 146 is 144.

Step: 2

$146$ lies between 12 and 16 on the number line.

[

$144$ = 12]

Step: 3

The perfect square nearer to 62 is 64.

Step: 4

$62$ lies between 4 and 8 on the number line.

[

$64$ = 8]

Step: 5

The perfect square nearer to 8 is 9.

Step: 6

$8$ lies between 0 and 4 on the number line.

[

$9$ = 3]

Step: 7

The perfect square nearer to 98 is 100.

Step: 8

$98$ lies between 8 and 12 on the number line.

[

$100$ = 10]

Step: 9

So, the point P represents

$98$ on the number line.

Correct Answer is :

$98$

Q13A square has an area of 16 square inches. If each side length were doubled, what would be the square's new area?

A. 32 square inches
B. 64 square inches
C. 16 square inches
D. 8 square inches

Solutions

Step: 1

Area of the square = side² = 16 square inches

[Given.]

Step: 2

Side = 4 inches

[Take square root on both sides of the equation and simplify.]

Step: 3

Double the length of the side = 2 × 4 inches = 8 inches

Step: 4

The area of the new square = 8 in. × 8 in. = 64 in.²

Correct Answer is : 64 square inches

Q14Identify the square root that the point represents on the number line.

A. -

$6$
B.

$6$
C. -

$8$
D.

$19$

Solutions

Step: 1

The point on the line lies between 0 and - 5 .

Step: 2

Among the choices -

$6$ , -

$8$ lie between 0 and - 5.

[The point lie left to the 0 on the number line.]

Step: 3

The point on the number line is closer to 0.

Step: 4

Among the choices the approximate value of -

$6$ is closer to 0.

Step: 5

So, the point on the number line represents -

$6$ .

Correct Answer is : -

$6$

Q15Find the square root of 900.
A. 30
B. 14
C. 288
D. 30,736

Solutions

Step: 1

Square root of 900 =

$900$

[Represent the square root using the radical symbol.]

Step: 2

$302$

[30² = 900.]

Step: 3

= 30

Step: 4

So, the square root of 900 is 30.

Correct Answer is : 30

Solved Examples and Worksheet for Representing Squares and Square Roots

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HighSchool Math