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Is there a really quick way of showing that:

$$\sqrt[3]{49-25\sqrt{2}}$$

Can be written in the form:

$$a+b\sqrt{2}$$

Is there a way to generalize which integers $a$ and $b$ can be rewritten in such a way?

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3  
Why do you think it is possible to write that cubic root in that way? –  DonAntonio May 7 at 3:45
    
@DonAntonio I was able to do the algebra out (ie setting the expression equal to a+bsqrt2 and setting parts equal to one another.) I was also given the hint that a and b are integers so I was able to do some factoring along the way. –  Jackson May 7 at 3:49
    
One way is to notice that the minimum polynomial has degree $2$, so there's at least hope for this to be true; if the degree was higher than $2$, it's not possible. –  user61527 May 7 at 3:51
    
Then what are you asking, @Jackson: how to do it "quickly"? Perhaps you show your work to know what'd be "quickly" in this case... –  DonAntonio May 7 at 3:52
1  
@DonAntonio I was wondering if there was some sort of characteristic of these numbers that guaranteed this outcome. I solved it the standard algebra way (45=a^3-6ab^2 and 29=3a^2b+2b^3). Since I was given in the problem that a and b were integers, I used the fact that 29 is prime to determine a and b. –  Jackson May 7 at 4:00

1 Answer 1

If we assume $\sqrt[3]{45 - 29 \sqrt{2}} = a + b \sqrt{2}$ where $a$ and $b$ are integers, then cubing both sides and expanding gives us $a^3 + 3 a^2 b \sqrt{2} + 6 a b^2 + 2 b^3 \sqrt{2}$, so we want to solve $$\eqalign{a^3 + 6 a b^2 &= 45\cr 3 a^2 b + 2 b^3 &= -29\cr}$$ Now $b$ divides $3 a^2 b + 2 b^3$ and $29$ is prime, so $b$ must be $\pm 1$. Moreover $b = +1$ would make $3 a^2 + 2 b^3 > 0$, so we must have $b = -1$. Then the second equation becomes $-3 a^2 - 2 = -29$ or $a^2 = 9$, so $a = \pm 3$. Finally the first equation becomes $\pm 27 \pm 18 = 45$ which is true for $a=+3$. Thus the answer is $3 - \sqrt{2}$.

EDIT: The problem keeps getting edited, but here are some more general considerations. For $\sqrt[3]{A + B \sqrt{m}} = a + b \sqrt{m}$ where $m$ is a square-free integer, you need $$ \eqalign{ a (a^2 + 3 m b^2) &= A\cr b (3 a^2 + m b^2) &= B\cr}$$ so $a$ divides $A$ and $b$ divides $B$. That reduces you to finitely many possibilities. To cut it down further you might note that $a \equiv A \mod 3$ and $mb\equiv B \mod 3$, and (if $m > 0$) $a$ and $b$ have the same signs as $A$ and $B$ respectively. Also, $$ A^2 - B^2 m = (a^2 - b^2 m)^3$$ so a necessary condition for this to work is that $A^2 - B^2 m$ is the cube of an integer. For example, $49^2 - 2 \times 25^2 = 1151 $ is not the cube of an integer, so $49 - 25 \sqrt{2}$ won't work.

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The OP already solved it that way, see the comments to the question. He's asking about generalizations. –  Bill Dubuque May 7 at 4:05
    
I see the OP has now changed the question to $45 + 25 \sqrt{2}$. This one does not simplify. –  Robert Israel May 7 at 4:06
    
@Robert Sorry, I think somebody else mistakenly edited it. Good catch. –  Jackson May 7 at 4:07
    
... and now it's changed again, to $49 - 25 \sqrt{2}$. –  Robert Israel May 7 at 4:09
    
@RobertIsrael Yes I fixed it back. At some point, somebody even edited to say a+sqrtb, which is clearly wrong. –  Jackson May 7 at 4:10

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