MULTIPLYING RADICAL EXPRESSIONS

Example 1 :

Evaluate :

(√6)(√6) 

Solution :

= √6 ⋅ √6 

= √(6 ⋅ 6)

= 6

Example 2 :

Evaluate :

(3√12)(√6)

Solution :

= 3√12 ⋅ √6

= 3√(12 ⋅ 6) 

= 3√72

= 3√(6 ⋅ 6 ⋅ 2)

= 3(6√2)

= 18√2

Example 3 :

Evaluate :

√6(√3 + √12)

Solution :

= √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

Example 4 :

Evaluate :

(√2 + √7) (√5 - √6)

Solution :

= (√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

Example 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

Example 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

Example 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

Example 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

Example 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

Example 10 :

Evaluate :

(√15x²)(√10x³)

Solution :

= (√15x²)(√10x³)

= √(15x² ⋅ 10x³)

= √(5 ⋅ 3 ⋅ 2 ⋅ 5 ⋅ x² ⋅ x² ⋅ x)

= 5x²√(3 ⋅ 2 ⋅ x)

= 5x²√6x

Example 11 :

Evaluate :

(³√24)(³√9)

Solution :

= (³√24)(³√9)

= ³√(24 ⋅ 9)

= ³√(2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 3)

= 2 ⋅ 3

= 6

Example 12 :

Evaluate :

(³√250x³)(³√4y⁶)

Solution :

= (³√250x³)(³√4y⁶)

= ³√(250x³ ⋅ 4y⁶)

= ³√(1000x³y⁶)

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

= 10xy²

Example 13 :

Simplify the expression :

√2(√45 + √5)

Solution :

= √2(√45 + √5)

Using distributive property,

= √2 √45 + √2 √5

= √(2 x 45) + √(2 x 5)

= √(2 x 3 x 3 x 5) + √(2 x 5)

= 3 √(2 x 5) + √(2 x 5)

= 3 √10 + √10

= 4 √10

Example 14 :

Simplify the expression :

√3(√72 - 3√2)

Solution :

= √3(√72 - 3√2)

= √3 √72 - 3 √3√2

= √(3 x 72) - 3 √(3 x 2)

= √(3 x 6 x 6 x 2) - 3 √(3 x 2)

= 6√(3 x 2) - 3 √(3 x 2)

= 6√6 - 3 √6

Subtracting it, we get

= 3√6

Example 15 :

Simplify the expression :

√5(2√6x - √96x)

Solution :

= √5(2√6x - √96x)

= 2√5 √6x - √5 √96x

= 2√(5 ⋅ 6 ⋅ x) - √(5 ⋅ 4 ⋅ 4 ⋅ 6 ⋅ x)

= 2√30x - 4√30x

= -2√30x

Example 16 :

Simplify the expression :

√7y (√27y + 5√12y)

Solution :

= √7y (√27y + 5√12y)

= √7y √27y + √7y (5√12y)

= √7y √(3 ⋅ 3 ⋅  3 ⋅ y) + √7y (5√(2 ⋅ 2 ⋅ 3 ⋅ y)

= √7y 3√3 y + √7y (2 ⋅ 5√3y)

= 3 √[7y(3y)] + 10 √[7y(3y)]

= 3 y √21 + 10y √21

= 13√21

Example 17 :

Simplify the expression :

(4√2 - √98)²

Solution :

(4√2 - √98)²

Using the algebraic identity, we get

(a - b)² = a² - 2ab + b²

a = 4√2 and b = √98

= (4√2)² - 2(4√2) √98 + √98²

= 16 (2) - 8√(2 x 98) + 98

= 32 + 98 - 8√(2 x 7 x 7 x 2)

= 130 - 8 x 7 x 2

= 130 - 112

= 18

Example 18 :

Simplify the expression :

(√3 + √48)(√20 - √5)

Solution :

(√3 + √48)(√20 - √5)

Using the distributive property, we get

= √3√20 - √3√5 + √48 √20 - √48 √5

= √(3 x 20) - √(3 x 5) + √(48 x 20) - √(48 x 5)

= √(3 x 2 x 2 x 5) - √15 + √(4 x 4 x 3 x 4 x 5) - √(4 x 4 x 3 x 5)

= 2√15 - √15 + 4x2√(3 x 5) - 4√(3 x 5)

= 2√15 - √15 + 8√15 - 4√15

= 10√15 - 5√15

= 5√15

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