A Radical Equation #maths #exponential #education

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All solutions:

x^3 = 625
Subtract 625 from both sides

x^3 - 625 = 0
This is in difference of cubes form, we can factor it as (a - b)(a^2 + ab + b^2)

a = x, b = 5cbrt5
(x - 5cbrt5)(x^2 + 5cbrt5x (5cbrt5)^2)

Solve for each case:

Case 1: (real)

x - 5cbrt5 = 0
Add 5cbrt5 to both sides

x = 5cbrt5

Case 2: (Complex)

x^2 + (5cbrt5)x + (5cbrt5)^2 = 0
Use the quadratic formula

x = (-(5cbrt5) +- sqrt((5cbrt5)^2 -4(1)(5cbrt5)^2))/2(1)
Simplify the denominator, simplify the second term in the radical, and distribute the negative in the first term

x = (-5cbrt5 +- sqrt((5cbrt5)^2 - 4(5cbrt5)^2))/2
simplify the radical

x = (-5cbrt5 +- sqrt(- 3(5cbrt5)^2))/2
sqrt the -1 in the radical

x = (-5cbrt5 +- isqrt(3(5cbrt5)^2))/2
Split the radical into 2 parts

x = (-5cbrt5 +- (isqrt3)sqrt(5cbrt5)^2)/2
sqrt the (5cbrt5)^2
x = (-5cbrt5 +- (isqrt3)(5cbrt5))/2

x = 5cbrt5, (-5cbrt5 - (isqrt3)(5cbrt5))/2, and (-5cbrt5 + (isqrt3)(5cbrt5))/2

This is the solution that Wolfram Alpha presented. However, it looks a little different since they split the fraction and seperated the radicals and other parts away from eachother. It is still the same solution.

just_another_person_who_li
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x√x = 25
x^(3/2) = 25
x = 25^(2/3) = 625^(1/3)

cosmolbfu
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(X^1/2 X)^2 = 625, X = 625^1/3 = 5 X 5^1/3

diegosimonetti