Finding Rational numbers

Question

Please help with the following question:

Find rational numbers a and b such that:

$$\left(7 + 5\sqrt2\right)^{\frac13} = a + b \sqrt2$$

Thank you

Answer 1

check this:

$\sqrt2 b+a = ({7+5 \sqrt2)})^{\frac{1}{3}}$

Isolate terms with b to the left hand side. Subtract a from both sides:

$\sqrt2 b = ({7+5 \sqrt2)})^{\frac{1}{3}}-a$

Solve for b. Divide both sides by sqrt(2):

$b = \frac{({7+5 \sqrt2)})^{\frac{1}{3}}- a}{\sqrt2}$

Can you take it from here?

Answer 2

Note that

$$\sqrt2 b+a = ({7+5 \sqrt2)})^{\frac{1}{3}}\\ \implies (\sqrt2 b+a)^3 = 7+5 \sqrt{2}\\ \implies a^3 + 3b\sqrt{2}a^2 + 6b^2a + 2b^3\sqrt{2} = 7+5 \sqrt{2}\\ \implies (a^3+6b^2a) + (3ba^2+2b^3)\sqrt{2} = 7+5 \sqrt{2}\\ $$

This yields two equations:

$$a^3+6b^2a = 7\\ 3ba^2+2b^3 = 5$$

There is an immediate, obvious choice for $a$ and $b$ here.

Answer 3

Hint: Cubing both sides you have $$7+5\sqrt{2}=(a^3+6ab^2)+(2b^3+3a^2b)\sqrt{2}$$ Since $a,b$ are rational, equating the rational and irrational parts of both sides, you get $$a^3+6ab^2=7\\ 2b^3+3a^2b=5$$ Now, from the first equation you get $$b^2=\frac{7-a^3}{6a}$$. Put that in the second equation to get $$b^2(2b^2+3a^2)^2=25\\ \Rightarrow (7-a^3)\left(\frac{7-a^3}{3a}+3a^2\right)^2=150a\\ \Rightarrow (7-a^3)\left(8a^3+7\right)^2=9\cdot150\cdot a^3$$Now, let $a^3=x$ Then you need to solve the cubic equation $$(7-x)(8x+7)^2=1350x\\ \Rightarrow (7-x)(64x^2+112x+49)=1350x\\ \Rightarrow 64x^3-336x^2+615x-343=0$$ Amazingly, Wolfram alpha gives one real solution to this $ x=1$! which corresponds to the real solution of $a=1$ and hence $b=1$ which are obviously rational and hence you are done.