MULTIPLYING RADICAL EXPRESSIONS WORKSHEET

Question 1 :

Evaluate :

(√6)(√6) 

1. Answer :

= √6 ⋅ √6 

= √(6 ⋅ 6)

= 6

Question 2 :

Evaluate :

(3√12)(√6)

2. Answer :

= 3√12 ⋅ √6

= 3√(12 ⋅ 6) 

= 3√72

= 3√(6 ⋅ 6 ⋅ 2)

= 3(6√2)

= 18√2

Question 3 :

Evaluate :

√6(√3 + √12)

3. Answer :

= √6(√3 + √12)

= √6√3 + √6√12

= √(6 ⋅ 3) + √(6 ⋅ 12)

= √18 + √72

= √(3 ⋅ 3 ⋅ 2) + √(6 ⋅ 6 ⋅ 2)

= 3√2 + 6√2

= 9√2

Question 4 :

Evaluate :

(√2 + √7) (√5 - √6)

4. Answer :

= (√2 + √7) (√5 - √6)

= √2√5 + √2(-√6) + √7√5 + √7(-√6)

= √2√5 - √2√6) + √7√5 - √7√6

= √(2 ⋅ 5) - √(2 ⋅ 6) + √(7 ⋅ 5) - √(7 ⋅ 6)

= √10 - √12 + √35 - √42

= √10 - √(2 ⋅ 2 ⋅ 3) + √35 - √42

= √10 - 2√3 + √35 - √42

Question 5 :

Evaluate :

(2 - 3√5)(5 - √5)

Solution :

= (2 - 3√5)(5 - √5)

= 2(5) + 2(-√5) - 3√5(5) - 3√5(-√5)

= 10 - 2√5) - 15√5 + 3√(5 ⋅ 5)

= 10 - 17√5 + 3(5)

= 10 - 17√5 + 15

= 25 - 17√5

Question 6 :

Evaluate :

(5 + 4√3)(3 + √3)

Solution :

= (5 + 4√3) (3 + √3)

= 5(3) + 5(√3) + 4√3(3) + 4√3(√3)

= 15 + 5√3 + 12√3 + 4√(3 ⋅ 3)

= 15 + 17√3 + 4(3)

= 15 + 17√3 + 12

= 27 + 17√3

Question 7 :

Evaluate :

(5√24 + √3)(2 - √3)

Solution :

= (5√24 + √3)(2 - √3)

= (5√(2 ⋅ 2 ⋅ 2 ⋅ 3) + √3)(2 - √3)

= (10√(2 ⋅ 3) + √3)(2 - √3)

= (10√6 + √3)(2 - √3)

= 10√6(2) + 10√6(-√3) + √3(2) + √3(-√3)

= 20√6 - 10√(6 ⋅ 3) + 2√3 - √(3 ⋅ 3)

= 20√6 - 10√(2 ⋅ 3 ⋅ 3) + 2√3 - 3

= 20√6 - 10(3√(2) + 2√3 - 3

= 20√6 - 30√2 + 2√3 - 3

Question 8 :

Evaluate :

(-3√3x + 4)(√3x - 5)

Solution :

= (-3√3x + 4)(√3x - 5)

= -3√3x(√3x) - 3√3x(-5) + 4(√3x) + 4(-5)

= -3√(3x ⋅ 3x) + 15√3x + 4√3x - 20

= -3(3x) + 19√3x - 20

= -9x + 19√3x - 20

Question 9 :

Evaluate :

(-4√28x)(√7x³)

Solution :

= (-4√28x)(√7x³)

= -4√(28x ⋅ 7x³)

= -4√(28x ⋅ 7x³)

= -4√(7 ⋅ 2 ⋅ 2 ⋅ 7 ⋅ x² ⋅ x²)

= -4(7 ⋅ 2 ⋅ x²)

= -4(14x²)

= -56x²

Question 10 :

Evaluate :

(√15x²)(√10x³)

Solution :

= (√15x²)(√10x³)

= √(15x² ⋅ 10x³)

= √(5 ⋅ 3 ⋅ 2 ⋅ 5 ⋅ x² ⋅ x² ⋅ x)

= 5x²√(3 ⋅ 2 ⋅ x)

= 5x²√6x

Question 11 :

Evaluate :

(³√24)(³√9)

Solution :

= (³√24)(³√9)

= ³√(24 ⋅ 9)

= ³√(2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 3)

= 2 ⋅ 3

= 6

Question 12 :

Evaluate :

(³√50x³)(³√4y⁶)

Solution :

= (³√250x³)(³√4y⁶)

= ³√(250x³ ⋅ 4y⁶)

= ³√(1000x³y⁶)

= ³√(10 ⋅ 10 ⋅ 10 ⋅ x ⋅ x ⋅ x ⋅ y² ⋅ y² ⋅ y²)

= 10xy²

Question 13 :

The time t (in seconds) it takes an object to hit the ground is given by t = √(h/16), where h is the height (in feet) from which the object was dropped. 

a. How long does it take an earring to hit the ground when it falls from the roof of the building?

b. How much sooner does the earring hit the ground when it is dropped from two stories (22 feet) below the roof?

Solution :

t = √(h/16)

a)

Distance to be covered = 55 ft

t = √(55/16)

= √3.4375

t = 1.85

So, the required time taken is 1.85 seconds

b)  The distance to be covered = 55 - 22

= 33 ft

t = √(33/16)

= √2.0625

t = 1.43

Rounding to the nearest hundredth, the time is about 1.44 seconds. To find how much sooner the earring hits the ground, we calculate the difference in time

= 1.85 - 1.43

= 0.42 seconds

Rounding to the nearest hundredth, the difference in time is about 0.42 seconds. 

Question 14 :

Multiply (√3 + x) (√3 - x)

Solution :

(√3 + x) (√3 - x)

The given expression looks like (a + b) (a - b) = a² - b²

Here a = √3 and b = x

= (√3)² - x²

= 3 -  x²

Question 15 :

Find the area of the rectangle given below.

Solution :

Length of rectangle = 3√2 ft and width = √40 ft

Area of rectangle = length x width

= 3√2 x √40

= 3√(2 x 40)

= 3√(2 x 2 x 2 x 2 x 5)

= 3 x 2 x 2 √5

= 12 √5 square feet

So, area of the rectangle is 12 √5 square feet.

Question 16 :

Find the area of the triangle given below.

Solution :

Base = 10 √12 m, height = 6√2 m

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

= (1/2) x 10 √12 x 6 √2 

= 5√12 x 6√2

= 5√(2 x 2 x 3) x 6√2

= 5 x 6 √(2 x 2 x 3) x √2

= 30 √(2 x 2 x 2 x 3)

= 30 x 2 √6

= 60√6 square meter

So, area of triangle is 60√6 square meter.

Question 17 :

Find the area of the triangle given below.

Solution :

Base = 6 √12 in, height = 3√5 in

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

= (1/2) x 6 √12 x 3 √5

= 3 √12 x 3 √5

= 9 √(2 x 2 x 3 x 5)

= 9 x 2  √15

= 18 √15

Area of triangle is 18 √15 square inches.

Question 18 :

Find the area of the triangle given below.

Solution :

Base = 7 √6 yard, height = 2√18 yard

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

= (1/2) x 7 √6 x 2 √18

= 7 √6 x √18

= 7 √(6 x 6 x 3)

= 7 x 6  √3

= 42 √3

So, area of parallelogram is 42 √3 square yards.

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