SIMPLEST RADICAL FORM

Problem 1 :

What is √45 in simplest form?

a. 5√3     b. 9√5     c. 3√5     d. 15√3

Solution :

√45 = √(9 × 5)

= √9 × √5

= 3√5

So, option (c) is correct.

Problem 2 :

What is √90x⁵ y⁴ in simplest radical form?

a. 10x²y² √9x       b. 9x³y² √10x²y²

c. 3x²y² √10x        d. 18x⁴y³ √5xy

Solution :

√90x⁵ y⁴ = √ (9 × 10xx⁴y⁴)

= √ (3² x⁴y⁴ 10x)

= √3² √x⁴ √y⁴ √10x

= 3x²y² √10x

So, option (c) is correct.

Problem 3 :

Which radical expression when expressed in simplified form is 3√13?

a. √16     b. √39      c. √117      d. √507

Solution :

3√13 = 3 × √13

= √39

So, option (b) is correct.

Problem 4 :

Which value could replace the? to make the expression true?

√12a⁶b³ = 2a³b√?

a. 6b²       b. 3b      c. 4a³b²     d. 3a²b

Solution :

= √12a⁶b³

= √(4 ∙ 3a⁶b²b)

= √(2² ∙ b²a⁶ ∙ 3b)

= √2² ∙ √b² ∙ √a⁶ ∙ √3b

= 2a³b ∙ √3b

So, option (b) is correct.

Problem 5 :

For what value of x does ∛x simplify to 4∛5?

a. 20     b. 80      c. 320     d. 460

Solution :

∛x = 4∛5

x = (4∛5)³

x = 4³ (∛5)³

x = 64 × 5

x = 320

So, option (c) is correct.

Problem 6 :

If y³ = -56, what is the value of y?

a. -2∛14    b. -8∛7     c. -2∛7     d. -7∛8

Solution :

y³ = -56

y = ∛ (-56)

y = -∛ (8 × 7)

y = - ∛8 ∛7

y = -2∛7

So, option (c) is correct.

Problem 7 :

Write the radical expression ∛-216 in simplest form.

Solution :

= ∛-216

= ∛-6 ∙ -6 ∙ -6

= -6

Problem 8 :

Which value could replace the? To make expression true?

∛486 = 3∛?

Solution :

∛486 = 3∛x

= ∛ (3³ ∙ 18)

= ∛ (3³) ∙ ∛18

= 3∛18

Problem 9 :

∛24 - 7√3

Solution :

= ∛24 - 7√3

= ∛ (2³ ∙ 3) - 7√3

= ∛ 2³ ∙ ∛3 - 7√3

= 2∛3 - 7√3

Problem 10 :

8√2 + 3√8

Solution :

= 8√2 + 3√8

= 8√2 + 3 √ (2² ∙ 2)

= 8√2 + 3 (√2² ∙ √2)

= 8√2 + 3(2 ∙ √2)

= 8√2 + 6√2

= 14√2

Problem 11 :

(-5√36) (3√2)

Solution :

= (-5√36) (3√2)

= (-5 × 6) (3√2)

= -30 × 3√2

= -90√2

Problem 12 :

(3a²√3) (a√12)

a. 3a³√15    b. 18a³    c. 4a⁴√12    d. 9a³√2

Solution :

= (3a²√3) (a√12)

= (3a²√3) (a√ (4 ∙ 3))

= (3a²√3) (2a√3)

= 6a³ (√3 ∙ √3)

= 6a³ (3)

= 18a³

So, option (b) is correct.

Problem 13 :

The distance d (in miles) that you can see to the horizon with your eye level h feet above the water is given by

d = √(3h/2)

How far can you see when your eye level is 5 feet above the water?

Solution :

d = √(3h/2)

When h = 5

d = √[3(5)/2]

= √15/√2

To rationalize the denominator, we have to multiply both numerator and denominator by √2

= (√15/√2)(√2/√2)

= √15√2/2

= √30/2

Problem 14 :

Evaluate the function for the given value of x.

a) h(x) = √5x, x = 10

b) g(x) = √3x, x = 60

c) r(x) = √[3x/3x² + 6)], x = 4

Solution :

a) h(x) = √5x, x = 10

applying the value of x, we get

h(10) = √5(10)

= √(5 ⋅ 2 ⋅ 5)

= 5√2

b) g(x) = √3x, x = 60

applying the values of x, we get

g(60) = √3(60)

= √3 ⋅ 12 ⋅ 5

= √(3 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 5)

= 2⋅3√5

= 6√5

c) r(x) = √[3x/3x² + 6)], x = 4

By applying the value of x, we get

r(4) = √[3(4)/(3(4)² + 6)]

= √[12/(48 + 6)]

= √[12/54]

= √[2/9]

= √2/√(3 ⋅ 3)

= √2/3

Problem 15 :

What are the perimeter and area of a rectangle with length of 5√3 centimeters and width of 3√2 centimeters?

Solution :

Length of rectangle = 5√3 cm

Width of the rectangle = 3√2 cm

Area of rectangle = length (width)

=  5√3 ⋅ 3√2

= 15√(3 ⋅ 2)

= 15√6 cm²

Perimeter of rectangle = 2(length + width)

= 2(5√3 + 3√2) cm

Problem 16 :

What are the perimeter and area of a rectangle with length of 2√6 centimeters and width of √3 centimeters?

Solution :

Length of rectangle = 2√6 cm

Width of the rectangle = √3 cm

Area of rectangle = length (width)

=  2√6 ⋅ √3

= 2√(6 ⋅ 3)

= 2√(2 ⋅ 3 ⋅ 3)

= 2(3) √2

= 6√2 cm²

Perimeter of rectangle = 2(length + width)

= 2(2√6 + √3) cm

Problem 17 :

If the base of a triangle measures 6√2 meters and the height measures 3√2 meters, then what is the area?

Solution :

Base of triangle = 6√2 m

height = 3√2 m

Area of triangle = (1/2) ⋅ base ⋅ height

= (1/2) 6√2 (3√2)

= 3√2 (3√2)

= 9(2)

= 18 m²