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segfault

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Registered: 2019-01-10

Posts: 2

Express ∛(7 + 5√2) in the form x + y√2

The expression given is ∛(7 + 5√2), which is to be expressed in the form x + y√2. The answer given in the back of the book is 1 + √2, which is indeed numerically the same as ∛(7 + 5√2), but I'm damned if I can see how you get from one to the other. Any suggestions? Thanks in advance!

zetafunc

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Registered: 2014-05-21

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Re: Express ∛(7 + 5√2) in the form x + y√2

Hi segfault,

Welcome to the forum!

Suppose that there are some values x and y for which ∛(7 + 5√2) = x + y√2. What happens if you cube both sides of that equation?

segfault

Member

Registered: 2019-01-10

Posts: 2

Re: Express ∛(7 + 5√2) in the form x + y√2

Hi zetafunc,

I see where you're going with this, expanding (x + y√2)^3 gives x^3 + 3√2x^2y + 6xy^2 + 2√2y^3 = 7 + 5√2

Now if I substitute x = 1 and y = 1 on the LHS I do get the RHS, and this gives 1 + √2 as required,  but is there a more systematic way to find the values of x and y?
It reminds me of the technique of using undetermined coefficients in partial fractions, but not sure how to do it in this case...
Thanks again.

Last edited by segfault (2019-01-10 22:20:47)