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 :
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} = 5^{3}
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} = 5^{4}
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} = 8^{4}
x^{3} = 8^{4}
x = ∛8^{4}
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.
x^{4} = 16^{3}
x = ∜16^{3}
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} = 27^{2}
Take cube root on both sides.
∛(x-3)^{3 }= ∛27^{2}
∛(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.
x^{5} = (-32)^{3}
Take 5th root on both sides.
x = ^{5}√(-32)^{3}
x = ^{5}√((-2)^{5})^{3}
x = ^{5}√(-2)^{1}^{5}
x = ((-2)^{1}^{5})^{1/5}
x = (-2)^{3}
x = -8
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