MULTIPLYING BINOMIALS WITH RADICALS

Expand and simplify :

Problem 1 :

(4 + √3) (4 - √3)

Solution :

(4 + √3) (4 - √3)

= 4(4) - 4√3 + 4√3 - (√3)²

= 16 - 3

= 13

Problem 2 :

(5 - √2) (5 + √2)

Solution :

(5 - √2) (5 + √2)

= 5(5) + 5√2 - 5√2 - (√2)²

= 25 - 2

= 23

Problem 3 :

(√5 - 2 ) (√5 + 2)

Solution :

(√5 - 2) (√5 + 2)

= (√5)² + 2√5 - 2√5 - 4

= 5 - 4

= 1

Problem 4 :

(√7 + 4 ) (√7 - 4)

Solution :

(√7 + 4) (√7 - 4)

= (√7)² - 4√7 + 4√7 - 16

= 7 - 16

= -9

Problem 5 :

(3√2 + 2 ) (3√2 - 2)

Solution :

(3√2 + 2) (3√2 - 2)

= 3²(√2)² - 6√2 + 6√2 - 4

= 18 - 4

= 14

Problem 6 :

(2√5 - 1 ) (2√5 + 1)

Solution :

(2√5 - 1) (2√5 + 1)

= 2²(√5)² + 2√5 - 2√5 - 1

= 20 - 1

= 19

Problem 7 :

(5 - 3√3) (5 + 3√3)

Solution :

(5 - 3√3) (5 + 3√3)

= 5(5) + 15√3 - 15√3 - 9(√3)²

= 25 - 27

= -2

Problem 8 :

(2 - 4√2) (2 + 4√2)

Solution :

(2 - 4√2) (2 + 4√2)

= 2(2) + 8√2 - 8√2 - 16(√2)²

= 4 - 32

= -28

Problem 9 :

(1 + 5√7) (1 - 5√7)

Solution :

(1 + 5√7) (1 - 5√7)

= 1(1) - 5√7 + 5√7 - 25(√7)²

= 1 - 175

= -174

Problem 10 :

Find the area of the rectangle shown in the figure

Solution :

Area of rectangle = length x width

Length = √11 and width = √3

Area of rectangle = √11 (√3)

= √(11 x 3)

= √33 square units.

Problem 11 :

Find the area of the rectangle shown in the figure

Solution :

By looking at the measures, they are the same it must be a square

Area of square = side (side)

Side length = √3 + √5

= (√3 + √5) (√3 + √5)

= √3² + √3√5 + √5√3 + √5²

= √3² + √3√5 +  √5√3 + √5²

= 3 + √15 + √15 + 5

= 8 + 2√15

Problem 12 :

Complete the statement √2 √5 = √10 because

Solution :

√2 √5 = √10

Using the properties of radicals, when two quantities are multiplied then the product of them will be equal to them separately will be equal to the square root of product of those two.

√a x √b = √(a x b)

√a x √b = √ab

So, the given statement is true.

Problem 13 :

Complete the statement 2√3 + 5√3 = 7√3 but 7√3 + 3√5 ≠ 10√8

Solution :

2√3 + 5√3 = 7√3

Here these are like radicals, we may add them. Considering 7√3 and 3√5 the numerals inside the square root are not the same, so they are not like radicals, we cannot add them.

7√3 + 3√5 ≠ 10√8

So, we cannot add them.

Problem 14 :

The radius r of a sphere is given by

r = ³ √(3/4π) V

where V is the volume of the sphere. Estimate the volume of a spherical head of brain coral with a radius of 1.5 feet.

Solution :

r = ³ √(3/4π) V

Here r = 1.5 feet

1.5 = ³ √(3/4π) V

Raising power 3 on both sides, we get

(1.5)³ = (3/4π) V

V = 3.375(4π/3)

V = 14.13

Problem 15 :

The mean sustained wind velocity (in meters per second) of a hurricane is modeled by

v( p) = 6.3 √(1013 − p)

where p is the air pressure (in millibars) at the center of the hurricane. Estimate the air pressure at the center of the hurricane when the mean sustained wind velocity is 54.5 meters per second.

Solution :

v(p) = 6.3 √(1013 − p)

v(p) = 54.5 meter per second

54.5 = 6.3 √(1013 − p)

√(1013 − p) = 54.5/6.3

√(1013 − p) = 8.65

1013 - p = (8.65)²

1013 - p = 74.82

p = 1013 - 74.82

p = 938.17

Problem 16 :

The maximum speed v (in meters per second) of a trapeze artist is represented by

v = √2gh

where g is the acceleration due to gravity (g ≈ 9.8 m/sec²) and h is the height (in meters) of the swing path. Find the height of the swing path for a performer whose maximum speed is 7 meters per second.

Solution :

v = √2gh

Here g = 9.8 and v = 7

7 = √2(9.8)h

49 = 19.6h

h = 49/19.6

h = 2.5