SOLVE WITH RATIONAL EXPONENTS

Solve the rational exponent problems of x.

Example 1 :

4x^1/3 = 20

Solution :

4x^1/3 = 20

Divide by 4 on both sides.

x^1/3 = 5

Raise power 3 on both sides.

(x^1/3)³ = 5³

x = 125

Example 2 :

3x^1/4 = 15

Solution :

3x^1/4 = 15

Divide by 3 on both sides.

x^1/4 = 5

(x^1/4)⁴ = 5⁴

x = 625

Example 3 :

3x^3/4 = 24

Solution :

3x^3/4 = 24

Divide by 3 on both sides.

x^3/4 = 8

Raise power 4 on both sides.

(x^3/4)⁴ = 8⁴

x³ = 8⁴

x = ∛8⁴

x = ∛(8⋅8⋅8⋅8)

x = 8∛8

x = 8∛(2⋅2⋅2)

x = 8(2)

x = 16

Example 4 :

4x^1/3 + 20 = 0

Solution :

4x^1/3 + 20 = 0

Subtract 20 on both sides.

4x^1/3 = -20

Divide by 4 on both sides.

x^1/3 = -5

Raise power 3 on both sides.

x = (-5)³

x = -125

Example 5 :

x^4/3 - 16 = 0

Solution :

x^4/3 - 16 = 0

Add 16 on both sides.

x^4/3 = 16

Raise power 3 on both sides.

x⁴ = 16³

x = ∜16³

x = ∜(16⋅16⋅16)

x = ∜(4⋅4⋅4⋅4⋅4⋅4)

x = 4∜(2⋅2⋅2⋅2)

x = 4(2)

x = 8

Example 6 :

(x-3)^3/2 = 27

Solution :

(x-3)^3/2 = 27

Raise power on both sides.

(x-3)³ = 27²

Take cube root on both sides.

∛(x-3)³ = ∛27²

∛(x-3)³ = ∛(27⋅27)

(x-3) = ∛(3⋅3⋅3⋅3⋅3⋅3)

x-3 = 9

Add 3 on both sides.

x = 9+3

x = 12

Example 7 :

(x-7)⁴ = 16

Solution :

Take fourth root on both sides.

x-7 = ∜16

x-7 = ∜(2⋅2⋅2⋅2)

x-7 = 2

Add 7 on both sides.

x = 2+7

x = 9

Example 8 :

3x^5/3 + 96 = 0

Solution :

3x^5/3 + 96 = 0

Subtract 96 on both sides.

3x^5/3 = -96

Divide by 3 on both sides.

x^5/3 = -32

Raise power 3 on both sides.

x⁵ = (-32)³

Take 5th root on both sides.

x = ⁵√(-32)³

x = ⁵√((-2)⁵)³

x = ⁵√(-2)¹⁵

x = ((-2)¹⁵)^1/5

x = (-2)³

x = -8

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