SOLVE WITH RATIONAL EXPONENTS

To solve equations with rational exponents, we have to isolate the variable for which we have rational exponent.

If we have power 3/2, to get rid of 2 which is at the denominator, we have to raise power 2 on both sides. 

If we have power 5/3, to get rid of 3 which is at the denominator, we have to raise power 2 on both sides

Solve the rational exponent problems of x.

Example 1 :

4x1/3 = 20

Solution :

4x1/3 = 20

Divide by 4 on both sides.

x1/3 = 5

Raise power 3 on both sides.

(x1/3)3 = 53

x = 125

Example 2 :

3x1/4 = 15

Solution :

3x1/4 = 15

Divide by 3 on both sides.

x1/4 = 5

(x1/4)4 = 54

x = 625

Example 3 :

3x3/4 = 24

Solution :

3x3/4 = 24

Divide by 3 on both sides.

x3/4 = 8

Raise power 4 on both sides.

(x3/4)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 :

4x1/3 + 20 = 0

Solution :

4x1/3 + 20 = 0

Subtract 20 on both sides.

4x1/3 = -20

Divide by 4 on both sides.

x1/3 = -5

Raise power 3 on both sides.

x = (-5)3

x = -125

Example 5 :

x4/3 - 16 = 0

Solution :

x4/3 - 16 = 0

Add 16 on both sides.

x4/3 = 16

Raise power 3 on both sides.

x4 = 163

x = ∜163

x = ∜(16⋅16⋅16)

x = ∜(4⋅44⋅44⋅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)272

(x-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 :

3x5/3 + 96 = 0

Solution :

3x5/3 + 96 = 0

Subtract 96 on both sides.

3x5/3 = -96

Divide by 3 on both sides.

x5/3 = -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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