SIMPLIFYING RADICAL EXPRESSIONS WITH VARIABLES WORKSHEET

Problem 1 :

Simplify : 

√(16u⁴v³)

Problem 2 :

Simplify : 

√(147m³n³)

Problem 3 :

Simplify : 

³√(125p⁶q³)

Problem 4 :

Simplify : 

⁴√(x⁴/16)

Problem 5 :

Simplify : 

6√(72y²)

Problem 6 :

Simplify :

√(196a⁶b⁸c¹⁰)

Problem 7 :

Simplify :

√[(x + 11)² - 44x]

Problem 8 :

Simplify : 

√(121x⁸y⁶ ÷ 81x⁴y⁸)

Problem 9 :

Simplify : 

√(16x² - 24x + 9)

Problem 10 :

Simplify : 

√[x² + (1/x²) + 2]

Detailed Answer Key

Problem 1 :

Simplify : 

√(16u⁴v³)

Solution :

√(16u⁴v³)  =  √(4 ⋅ 4 ⋅ u² ⋅ u² ⋅ v ⋅ v ⋅ v)

√(16u⁴v³)  =  4u²v√v

Problem 2 :

Simplify : 

√(147m³n³)

Solution :

√(147m³n³)  =  √(7 ⋅ 7 ⋅ 3 ⋅ m ⋅ m ⋅ m ⋅ n ⋅ n ⋅ n)

√(147m³n³)  =  7mn√(3mn)

Problem 3 :

Simplify : 

³√(125p⁶q³)

Solution :

³√(125p⁶q³)  =  ³√(5 ⋅ 5 ⋅ 5 ⋅ p² ⋅ p² ⋅ p² ⋅ q ⋅ q ⋅ q)

³√(125p⁶q³)  =  5p²q

Problem 4 :

Simplify : 

⁴√(x⁴/16)

Solution :

⁴√(x⁴/16)  =  ⁴√(x⁴) / ⁴√16

⁴√(x⁴/16)  =  ⁴√(x ⋅ x ⋅ x ⋅ x) / ⁴√(2 ⋅ 2 ⋅ 2 ⋅ 2)

⁴√(x⁴/16)  =  x / 2

Problem 5 :

Simplify : 

6√(72y²)

Solution :

6√(72y²)  =  6√(6 ⋅ 6 ⋅ 2 ⋅ y ⋅ y)

6√(72y²)  =  6(6y)√2

6√(72y²)  =  12y√2

Problem 6 :

Simplify :

√(196a⁶b⁸c¹⁰)

Solution :

√(196a⁶b⁸c¹⁰)  =  √(14 ⋅ 14 ⋅ a³ ⋅ a³ ⋅ b⁴ ⋅ b⁴ ⋅ c⁵ ⋅ c⁵)

√(196a⁶b⁸c¹⁰)  =  14a³b⁴c⁵

Problem 7 :

Simplify :

√[(x + 11)² - 44x]

Solution :

√[(x + 11)² - 44x]  =  √[x² + 2(x)(11) + 11² - 44x]

√[(x + 11)² - 44x]  =  √[x² + 22x + 121 - 44x]

√[(x + 11)² - 44x]  =  √[x² - 22x + 121]

√[(x + 11)² - 44x]  =  √[(x - 11)(x - 11)]

√[(x + 11)² - 44x]  =  x - 11

Problem 8 :

Simplify : 

√(121x⁸y⁶ ÷ 81x⁴y⁸)

Solution : 

√(121x⁸y⁶ ÷ 81x⁴y⁸)  =  √(121x⁸y⁶ / 81x⁴y⁸)

√(121x⁸y⁶ ÷ 81x⁴y⁸)  =  √(121x^8-4 / 81y^8-6)

√(121x⁸y⁶ ÷ 81x⁴y⁸)  =  √(121x⁴ / 81y²)

√(121x⁸y⁶ ÷ 81x⁴y⁸)  =  √(11²x⁴ / 9²y²)

√(121x⁸y⁶ ÷ 81x⁴y⁸)  =  11x² / 9y

Problem 9 :

Simplify : 

√(16x² - 24x + 9)

Solution : 

√(16x² - 24x + 9)  =  √[4²x² - 2(4x)(3) + 3²]

√(16x² - 24x + 9)  =  √[(4x)² - 2(4x)(3) + 3²]

Using the algebraic identity (a - b)²  =  a² - 2ab + b² on the right side, 

√(16x² - 24x + 9)  =  √(4x - 3)²

√(16x² - 24x + 9)  =  (4x - 3)

Problem 10 :

Simplify : 

√[x² + (1/x²) + 2]

Solution : 

√[x² + (1/x²) + 2]  =  √[x² + 2 + (1/x²)]

√[x² + (1/x²) + 2]  =  √[x² + 2(x)(1/x) + (1/x)²]

Using the algebraic identity (a + b)²  =  a² + 2ab + b² on the right side, 

√[x² + (1/x²) + 2]  =  √(x² + 2(x)(1/x) + (1/x)²]

√[x² + (1/x²) + 2]  =  √(x + 1/2)²

√[x² + (1/x²) + 2]  =  x + 1/2

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