Solve for x:
[Sixth root of 27]+[cube root of x] = 6/(3-[square root of 3]
27^(1/6) + x^(1/3) = 6/(3 - √3)
(3^3)^(1/6) + x^(1/3) = 6/(3 - √3)[(3+√3)/(3+√3)]
3^(1/2) + x^(1/3) = 3 + 2√3
x^(1/3) = 3 + √3)
x = (3+√3)^3
To solve for x in the equation [Sixth root of 27] + [cube root of x] = 6/(3 - [square root of 3]), follow these steps:
Step 1: Simplify the equation
Start by evaluating all the roots and simplifying the right side of the equation.
The sixth root of 27 is equal to 3 because 3^6 = 729 and 3^5 = 243.
The cube root of x remains as it is.
The square root of 3 remains as it is.
The right side of the equation can be simplified as follows:
6/(3 - [square root of 3]) = 6/(3 - √3) = 6/[(3 - √3) * (3 + √3)] = 6/(9 - 3√3 + 3√3 - (√3)^2) = 6/(9 - 3) = 6/6 = 1
So, the simplified equation is 3 + cube root of x = 1.
Step 2: Isolate the unknown term
To get rid of the 3 on the left side, subtract 3 from both sides of the equation:
3 + cube root of x - 3 = 1 - 3
Simplifying, the equation becomes cube root of x = -2.
Step 3: Remove the cube root
To eliminate the cube root on the left side, cube both sides of the equation:
(cube root of x)^3 = (-2)^3
This simplifies to x = -8.
Therefore, the solution for x is -8.