SOLVING EQUATIONS WITH nth ROOTS

Find the real solution(s) of the following.

Problem 1 :

4x⁵ = 128

Solution :

4x⁵ = 128

Divide by 4 on both sides.

x⁵ = 128/4

x⁵ = 32

Taking 5th root on both sides, we get

5^th root (x) = 5^th root (32)

5^th root (x) = 5^th root (2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2)

= 2

So, the value of x is 2.

Problem 2 :

(x − 3)⁴ = 21

Solution :

(x − 3)⁴ = 21

Take 4^th root on both sides, we get

x - 3 =  ∜21

x - 3 = ±2.14

x - 3 = 2.14 and x - 3 = -2.14

x = 2.14 + 3 and x = -2.14 + 3

x = 5.14 and x = 0.86

Find the real solution(s) of the equation. Round your answer to two decimal places when appropriate.

Problem 3 :

8x³ = 64

Solution :

8x³ = 64

Dividing by 8 on both sides, we get

x³ = 64/8

x³ = 8

Taking cube roots on both sides, we get 

x =  ∛8

= ∛(2 ⋅ 2 ⋅ 2)

x = 2

Problem 4 :

(1/2) x⁵ = 512

Solution :

(1/2) x⁵ = 512

Multiplying by 2 on both sides, we get

x⁵ = 512(2)

x⁵ = 1024

Taking 5th root on both sides.

x = 5^th root (1024)

x = 5^th root (4 ⋅ 4 ⋅ 4 ⋅ 4 ⋅ 4)

x = 4

Problem 5 :

(x + 5)⁴ = 16

Solution :

(x + 5)⁴ = 16

Taking 4^th root on both sides, we get

(x + 5) = ∜16

(x + 5) = ∜(2 ⋅ 2 ⋅ 2 ⋅ 2)

x + 5 = ±2

x + 5 = 2 and x + 5 = -2

x = 2 - 5 and x = -2 - 5

x = -3 and x = -7

Problem 6 :

(x − 2)³ = −14

Solution :

(x − 2)³ = −14

Take cube root on both sides.

x - 2 = ∛-14

x - 2 = -2.41

Add 2 on both sides.

x = -2.41 + 2

x = -0.41

Find the radius of the figure with the given volume.

Problem 7 :

V = 216 ft³

Solution :

Volume of sphere = (4/3)πr³

(4/3)πr³ = 216

r³ = 216 (3/4) (1/π)

r³ = 51.59

r = 3.7 ft

Problem 8 :

Volume 1332 ft³

Solution :

Volume of sphere = πr² h

πr² h = 1332

πr² 9 = 1332

r² = (1332/9)(1/π)

r² = (1332/9)(1/π)

r = 6.86