SIMPLIFY RADICAL EXPRESSIONS WITH VARIABLES

Note :

The same rules to be followed, when we have variables.

Simplify the expressions. Assume all variables represent positive rational number.

Problem 1 :

∛y⁵/27y³

Solution :

∛y⁵ = ∛(y ∙ y ∙ y ∙ y ∙ y)

∛27y³ = ∛(3 ∙ 3 ∙ 3 ∙ y ∙ y ∙ y)

∛y⁵/27y³ = ∛ (y³ ∙ y²) / ∛ (3³ ∙ y³)

= ∛y² ∙ y / 3y

= ∛y²/3

Problem 2 :

∛16z³

Solution :

∛16z³ = ∛(2 ∙ 2 ∙ 2 ∙ 2 ∙ z ∙ z ∙ z)

= 2 ∛2 ∙ z

= 2z ∛2

Problem 3 :

∛16a³

Solution :

∛16a³ = ∛(2 ∙ 2 ∙ 2 ∙ 2 ∙ a ∙ a ∙ a)

= 2 ∛2 ∙ a

= 2a ∛2

Problem 4 :

∛b⁴/27b

Solution :

∛b⁴ = ∛(b ∙ b ∙ b ∙ b)

∛27b = ∛(3 ∙ 3 ∙ 3 ∙ b)

∛b⁴/27b = ∛ (b³ ∙ b) / ∛ (3³ ∙ b)

= b/3

Problem 5 :

∛x⁵/x²

Solution :

∛x⁵ = ∛(x ∙ x ∙ x ∙ x ∙ x)

∛x² = ∛(x ∙ x)

∛x⁵/x² = ∛ (x³ ∙ x²) / ∛ (x²)

= ∛ x³

= x

Problem 6 :

∛15m⁴n²²

Solution :

= ∛15m⁴n²²

= ∛ [(15 ∙ m ∙ m ∙ m ∙ m) (n³ ∙ n³ ∙ n³ ∙ n³ ∙ n³ ∙ n³ ∙ n³ ∙ n)]

∛15m⁴n²² = mn⁷ ∛15mn

Problem 7 :

∛y¹¹/y²

Solution :

∛y¹¹ = ∛y³ ∙ y³ ∙ y³ ∙ y ∙ y

∛y² = ∛y ∙ y

∛y¹¹/y² = ∛ (y³ ∙ y³ ∙ y³ ∙ y ∙ y) / ∛ (y ∙ y)

= y³

Problem 8 :

∛20s¹⁵ t¹¹

Solution :

= ∛20s¹⁵ t¹¹

= ∛ [(20 ∙ s³ ∙ s³ ∙ s³ ∙ s³ ∙ s³). (t³ ∙ t³ ∙ t³ ∙ t ∙ t)]

∛20s¹⁵ t¹¹ = s⁵ t³ ∛20t²

Problem 9 :

∛32/∛-4

Solution :

= ∛32/∛-4

= ∛(-32/4)

= ∛-8

= -2

Problem 10 :

∛162x⁵/∛3x²

Solution :

= ∛162x⁵/∛3x²

= ∛54x³

= ∛ (3³ ∙ 2x³)

= ∛3³ ∙ 2∙ x∙ x∙ x)

= 3x ∛2

= 3 ∛2x

Problem 11 :

√12x⁴/√3x

Solution :

= √12x⁴/√3x

= (√12x⁴ × √3) / (√3x × √3)

= √36x⁴/√9x

= 6x²/3x

= 2x

Problem 12 :

∛80n⁵

Solution :

= ∛80n⁵

= ∛ (2³ × 10n² × n³)

= ∛2³ × ∛10n² × ∛n³

= 2n ∛10n²

Problem 13 :

∛p¹⁷q¹⁸

Solution :

= ∛p¹⁷q¹⁸

= ∛ (p³ ∙ p³ ∙ p³ ∙ p³ ∙ p³ ∙ p²) (q³ ∙ q³ ∙ q³ ∙ q³ ∙ q³ ∙ q³)

= p⁵ q⁶ ∛p²

Problem 14 :

When 5 √20 is written in simplest radical form, the result is k√5. What is the value of k?

a)   20      b) 10      c) 7       d) 4

Solution :

= 5√20

Decomposing 20, we get √(2 x 2 x 5)

= 5√(2 x 2 x 5)

= 5 x 2 √5

= 10√5

k√5 = 10√5

Comparing the corresponding terms, we get the value of k is 10.

Problem 15 :

Which expression is equivalent to 7√90?

a) 16 √10     b) 21√10      c) 70 √9         d) 6√30

Solution :

= 7√90

Decomposing 90, we get

= 7√(2 x 3 x 3 x 5)

= 7 x 3√(2 x 5)

= 21√10

Problem 15 :

Multiply 4√3(2√3 - 3√6)

a) 16 √10     b) 21√10      c) 70 √9         d) 6√30

Solution :

= 4√3(2√3 - 3√6)

Distributing 4√3, we get

= 4√3(2√3) - 4√3(3√6)

= 8√3√3 - 12 √3 √6

= 8(3) - 12√(3 x 6)

= 24 - 12√(3 x 3 x 2)

= 24 - (12 x 3)√2

= 24 - 36√2

Problem 16 :

Multiply (3√x - √y)²

Solution :

= (3√x - √y)²

= (3√x)² - 2(3√x)√y + (√y)²

= 9x - 6√xy + y

Problem 17 :

Marcy received a text message from Mark asking her how old she was. In response, Marcy texted back “125^(2/3) years old.” Help Mark determine how old Marcy is.

Solution :

= 125^(2/3)

Decomposing 125, we get

= (5³)^2/3

= 5^{3 x (2/3)}

= 5²

= 25

Problem 18 :

(1/2) √8 + (3/5) √50 - (2/3) √18

Solution :

= (1/2) √8 + (3/5) √50 - (2/3) √18

= (1/2) √(2 x 2 x 2) + (3/5) √(5 x 5 x 2) - (2/3) √(3 x 3 x 2)

= (1/2) x 2√2 + (3/5) x 5√2 - (2/3) x 3√2

= √2 + 3√2 - 2√2

= 4√2 - 2√2

= 2√2