Expert: Paul Klarreich - 10/3/2008

Question
Hi,

I am stuck on the following question and I am unable to identify the type of problem it could be. Thank you for your help.
Verify z=( 4 + sqrt15) ^ 1/3 + (4- sqrt15) ^ 1/3 satisfies

z^3 -3z -8 =0

Answer
Questioner:   Clarissa
Category:  Advanced Math
Private:  No
 
Subject:  unsure - law of indices?
Question:  Hi,

I am stuck on the following question and I am unable to identify the type of problem it could be. Thank you for your help.
Verify z= (4 + sqrt(15))^1/3 + (4 - sqrt(15))^1/3 satisfies

z^3 -3z -8 =0
.....................................
Hi, Clarissa,

Alas there is no simple way of doing it.  There is a formula (look up Cardano) for solving cubic equations.  This formula is only about 3 pages long -- no wonder nobody uses it.

So I guess we just have to substitute and do the arithmetic.

WARNING: VIEW THIS DISCUSSION IN A FIXED-SIZE FONT, SUCH AS COURIER.

z=( 4 + sqrt(15))^1/3 + (4 - sqrt(15))^1/3
   --------a--------   ---------b--------

Binomial expansion:

(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3;

z^3 = 4 + sqrt(15) + 4 - sqrt(15) + 3(4 + sqrt(15))^2/3(4 - sqrt(15))^1/3 + 3(4 + sqrt(15))^1/3(4 - sqrt(15))^2/3

z^3 = 8 + 3(4 + sqrt(15))^1/3(16 - 15)^1/3 + 3(16 - 15)^1/3(4 - sqrt(15))^1/3

z^3 = 8 + 3(4 + sqrt(15))^1/3(1)^1/3 + 3(1)^1/3(4 - sqrt(15))^1/3

z^3 = 8 + 3(4 + sqrt(15))^1/3 + 3(4 - sqrt(15))^1/3

z^3 = 8 + 3[ (4 + sqrt(15))^1/3 + (4 - sqrt(15))^1/3 ]

As the Scooter would say, "HOLY COW!  That thing inside the brackets is your z!!!!"

z^3 = 8 + 3[                     z                   ]

If z^3 = 8 + 3z, then z^3 - 3z - 8 = 0

No kidding.