FIND SQUARE ROOT OF RATIONAL NUMBER

Evaluate each of the following.

Problem 1 :

√1/25

Solution :

= √1/25

= √1/√25

= √(1 ⋅ 1) / √(5 ⋅ 5)

= 1/5

Problem 2 :

√1/81

Solution :

= √1/81

= √1/√81

= √(1 ⋅ 1)/√(9 ⋅ 9)

= 1/9 

Problem 3 :

-√36/144

Solution :

= -√36/144

= -√36/√144

= -√(6 ⋅ 6)/√(12 ⋅ 12)

= -6/12

= -1/2

Problem 4 :

√529/625

Solution :

= √529/625

= √529/√625

= √(23 ⋅ 23)/√(25 ⋅ 25)

= 23/25

Problem 5 :

√16/25

Solution :

= √16/25

= √16/√25

= √(4 ⋅ 4)/√(5 ⋅ 5)

= 4/5

Determine whether each of the following statements in true or false.

Problem 6 :

-√36/324 = 1/3

Solution :

= -√36/324

= -√36/√324

= -√(6⋅6)/√(18⋅18)

= -6/18

= -1/3

So, the statement is false.

Problem 7 :

√324/625 = 18/25

Solution :

= √324/625

= √324/√625

= √(18⋅18)/√(25⋅25)

= 18/25

So, the statement is true.

Problem 8 :

-√900/961 = -30/31

Solution :

= -√900/961

= -√900/√961

= -30/31

So, the statement is true.

Problem 9 :

√1/4 = ± 1/2

Solution :

= √1/4

= √1/√4

= ± 1/2

So, the statement is true.

Problem 10 :

If x/y = -2, find √ (x²/y² + y²/x²)

Solution :

√ (x²/y² + y²/x²) = √ (x/y)² + (y/x)²

= √ (x/y)² + (x/y)^-²

= √ (-2)² + (-2)^-2

= √4 + 1/4

= √17/4

= √17/√4

= √17/2

Problem 11 :

What is the radius of Kingsley Lake? Use 3.14 for 𝛑

Kingsley Lake in Clay County, Florida is a circular lake that covers an area of about 8,038,400 square meters.

Solution :

Area of the lake = 8,038,400 square meters.

𝛑r² = 8,038,400

3.14 r² = 8,038,400

r² = 8,038,400 / 3.14

r² = 2560000

r = √2560000

r = √1600(1600)

= 1600

So, the radius is 1600 meters

Problem 12 :

What is the perimeter of a square with an area of 900 square feet?

Solution :

Area of square = 900 square feet

Let x be the side length of the square, then area = x²

x² = 900

x = √900

x = √30(30)

= 30

Perimeter of the square = 4x

= 4(30)

= 120 feet

So, the required perimeter is 120 feet.

Problem 13 :

What is the diameter of a circle with an are of 100π square yards?

Solution :

𝛑r² = 100π

r² = 100

r = √100

r = 10 yards

Diameter of the circle = 2(10)

= 20 yards.

Problem 14 :

Evaluate the expression :

a) 12 - 3√25

b) √(28/7) + 2.4

c) 5(√49 - 10)

Solution :

a) 12 - 3√25 = 12 - 3√(5 x 5)

= 12 - 3(5)

= 12 - 15

= -3

b) √(28/7) + 2.4

= √4 + 2.4

= √(2 x 2) + 2.4

= 2 + 2.4

= 4.4

c) 5(√49 - 10)

= 5(√7 x 7 - 10)

= 5(7 - 10)

= 5(-3)

= -15

Problem 15 :

The area of a circle is 2826 square feet. Write and solve an equation to find the radius of the circle. Use 3.14 for π.

Solution :

Area of the circle (𝛑r²) = 2826

Using 𝛑  = 3.14, we get

3.14r² = 2826

r² = 2826/3.14

r² = 900

r = √900

= √(30 x 30)

r = 30

So, the radius of the circle is 30 feet.

Problem 16 :

Is 26 a perfect square?

Solution :

A number which can be written as product of two same numbers is known as perfect square. Since 26 cannot be written as product of two same numbers, it cannot be considered as perfect square.

Problem 17 :

Can the square of an integer be a negative number?

Solution :

Square of positive and negative numbers will be positive only. So, square of integer cannot be negative.

Problem 18 :

Does √256 represent the positive square root of 256, the negative square root of 256, or both? Explain.

Solution :

√256 = √(16 x 16)

= -16 or 16

-√256 = -√(16 x 16)

= -16

So, both are the same.

Problem 19 :

The area of the base of a square notepad is 9 square inches. What is the length of one side of the base of the notepad?

Solution :

Area of the square = x²

x² = 9

x = √9

x = √(3 x 3)

= 3

So, the side length of the square is 3 inches.