simplify 1 over 1 minus cube root of radical 3

Question

When you multiply by the conjugate you still have 1 minus cube root of radical 9 in denominator. How do you get rid of radical in denominator?

Answer 1

You have a cube root in your denominator. Conjugates only apply to square roots and complex numbers! Don't use them with cube roots, etc.!

 

1

-------------

1 - cbrt(3)

 

What if you could make a difference of two cubes out of that denominator? After all, a cube root is the inverse of a cube.

 

a³ - b³ = (a - b)(a² + ab + b²)

 

You have your (a - b) factor: a = 1, b = cbrt(3). Just substitute 1 for a and cbrt(3) for b. Multiply your fraction by the (a² + ab + b²) factor over itself:

 

1

-------------

1 - cbrt(3)

 

times

 

1 + cbrt(3) + cbrt(9)

--------------------------

1 + cbrt(3) + cbrt(9)

 

=

 

1 + cbrt(3) + cbrt(9)

--------------------------

1 - 3

 

=

 

1 + cbrt(3) + cbrt(9)

--------------------------

-2