SOLVING EQUATIONS WITH RADICALS AND RATIONAL EXPONENTS

Solve each equation. 

Problem 1 :

0 = -2 + √(6x – 1)

Solution :

0 = -2 + √(6x – 1)

Add 2 on both sides.

2 = √(6x – 1)

Raise both sides to the power 2.

(2)² = (√(6x – 1))²

4 = 6x – 1

Add 1 on both sides.

5 = 6x

Divide both sides by 6.

x = 5/6

Problem 2 :

√(4x – 20) - √(12 - 2x) = 0

Solution :

√(4x – 20) - √(12 - 2x) = 0

√(4x – 20) = √(12 - 2x)

Raise both sides to the power 2.

(√(4x – 20))² = (√(12 - 2x))²

4x – 20 = 12 – 2x

4x + 2x = 12 + 20

6x = 32

Divide both sides by 6.

6x/6 = 32/6

x = 16/3

Problem 3 :

-6 + 6(26 – a)^4/3 = 90

Solution :

-6 + 6(26 – a)^4/3 = 90

Add 6 on both sides.

6(26 – a)^4/3 = 90 + 6

6(26 – a)^4/3 = 96

Divide both sides by 6.

(26 – a)^4/3= 96/6

(26 – a)^4/3= 16

Raise both sides to the power 3/4.

((26 – a)^4/3)^3/4 = (16)^3/4

26 – a = (∜16)³

26 – a = (∜2⁴)³

26 – a = (∜2¹²)

26 – a = 2³

26 – a = 8

-a = 8 – 26

-a = -18

a = 18

Problem 4 :

√(7a + 2) – 2 = 7a

Solution :

√(7a + 2) – 2 = 7a

Add 2 on both sides.

√(7a + 2) = 7a + 2

Raise both sides to the power 2.

(√(7a + 2))² = (7a + 2)²

7a + 2 = (7a)² + 2² + 2(7a)(2)

7a + 2 = 49a² + 4 + 28a

7a + 2 - 49a² - 4 - 28a = 0

-49a² - 21a - 2 = 0

49a² + 21a + 2 = 0

49a² + 7a + 14a + 2 = 0

(49a² + 7a) + (14a + 2) = 0

7a(7a + 1) + 2(7a + 1) = 0

(7a + 2) (7a + 1) = 0

7a + 2 = 0 and 7a + 1 = 0

7a = -2 7a = -1

a = -2/7 a = -1/7

Problem 5 :

(4x – 3)^3/2 = 8

Solution :

(4x – 3)^3/2 = 8

Raise both sides to the power 2/3.

((4x – 3)^3/2)^2/3 = (8)^2/3

4x – 3 = (2³)^2/3

4x – 3 = 4

Add 3 on both sides.

4x = 4 + 3

4x = 7

Divide both sides by 4.

x = 7/4

Problem 6 :

(-2 – x)^0.5 – x = 2

Solution :

(-2 – x)^0.5 – x = 2

Decimal number 0.5 convert to the fraction. 

(-2 – x)^5/10 – x = 2

(-2 – x)^1/2 – x = 2

Add x on both sides.

(-2 – x)^1/2 = 2 + x

Raise both sides to the power 2.

((-2 – x)^1/2)² = (2 + x)²

-2 – x = 2² + x² + 2(2)(x)

-2 – x = 4 + x² + 4x

-2 – x – 4 – x² – 4x = 0

-x² – 5x – 6 = 0

x² + 5x + 6 = 0

(x + 2) (x + 3) = 0

x + 2 = 0 and x + 3 = 0

x = -2 x = -3

Solve :

Problem 7 :

(x + 5)^2/3 = 4

Solution :

(x + 5)^2/3 = 4

Raise both sides to the power 3/2.

((x + 5)^2/3)^3/2 = (4)^3/2

x + 5 = (√4)³

x + 5 = (√2²)³

x + 5 = 2³

x + 5 = 8

Subtract 5 on both sides.

x = 8 – 5

x = 3

Problem 8 :

(x - 2)^2/3 = 9

Solution :

(x - 2)^2/3 = 9

Raise both sides to the power 3/2.

((x - 2)^2/3)^3/2 = (9)^3/2

x – 2 = (√9)³

x – 2 = ((√3)²)³

x – 2 = (√3)⁶

x – 2 = 3³

x – 2 = 27

x = 27 + 2

x = 29

Problem 9 :

3(x - 2)^3/4 = 24

Solution :

3(x - 2)^3/4 = 24

Divide both sides by 3.

(x - 2)^3/4 = 24/3

(x - 2)^3/4 = 8

Raise both sides to the power 4/3.

((x - 2)^3/4)^4/3 = (8)^4/3

x - 2 = (∛8)⁴

x – 2 = ∛(2³)⁴

x – 2 = ∛2¹²

x – 2 = 2⁴

x – 2 = 16

Add 2 on both sides.

x = 18

Problem 10 :

3(x + 3)^3/4 = 81

Solution :

3(x + 3)^3/4 = 81

Divide both sides by 3.

(x + 3)^3/4 = 81/3

(x + 3)^3/4 = 27

Raise both sides to the power 4/3.

((x + 3)^3/4)^4/3 = (27)^4/3

x + 3 = (∛27)⁴

x + 3 = ∛(3³)⁴

x + 3 = ∛3¹²

x + 3 = 3⁴

x + 3 = 81

Subtract 3 on both sides.

x = 81 - 3

x = 78

Problem 11 :

(x + 1)^3/2 - 2 = 25

Solution :

(x + 1)^3/2 - 2 = 25

Add 2 on both sides.

(x + 1)^3/2 = 25 + 2

(x + 1)^3/2 = 27

Raise both sides to the power 2/3.

((x + 1)^3/2)^2/3 = (27)^2/3

x + 1 = (∛27)²

x + 1 = (∛3³)²

x + 1 = 3²

x + 1 = 9

Subtract 1 on both sides.

x = 9 – 1

x = 8

Problem 12 :

3 + (4 - x)^3/2 = 11

Solution :

3 + (4 - x)^3/2 = 11

Subtract 3 on both sides.

(4 - x)^3/2 = 8

Raise both sides to the power 2/3.

((4 - x)^3/2)^2/3 = (8)^2/3

4 - x = (∛8)²

4 - x = (∛2³)²

4 - x = 2²

4 - x = 4

Subtract 4 on both sides.

x = 0