Phi with a and b

Algebra Level 4

The square root of phi can be expressed as the sixth root of the quantity root a plus b. What is a minus b?

In other words, $\sqrt{\phi}=\sqrt[6]{\sqrt{a}+b}$

The answer is 3.

3 solutions

Daniel Liu
Apr 26, 2014

We have $\varphi^3=\sqrt{5}+2$ so the answer is $5-2=\boxed{3}$ .

Exactly bro. :D

Finn Hulse - 7 years, 1 month ago

Nice solution, I thought it was much more difficult so i rated it level 4. Oops.

Nathan Ramesh - 7 years, 1 month ago

Why is $\phi^{3} = \sqrt5 + 2$ ?

Omkar Kulkarni - 6 years, 5 months ago

phi is defined as $\sqrt{5}+\frac{1}{2}$ Cubing this results in $\sqrt{5}+2$

Lawrence Mayne - 6 years, 2 months ago

$\phi=\dfrac{1+\sqrt5}{2}$ , which is the positive root of $x^2-x-1=0$ . But $x^2-x-1=0~\Rightarrow~x^2=x+1~\Rightarrow~x^3=x^2+x=2x+1$ .

So $\phi^3=2\phi+1=2+\sqrt5$ .

Laurent Shorts - 4 years, 3 months ago

Abdur Rehman Zahid
Nov 22, 2014

$\sqrt{\phi}=\sqrt[6]{\sqrt{a}+b}$ $(\phi)^3=\sqrt{a}+b$ $\sqrt{5}+2=\sqrt{a}+b$ So $a=5,b=2$ and $a-b=5-2=\boxed{3}$

Finn Hulse
Apr 26, 2014

Nice problem Nathan. :D

:D people are telling me it's pronounced fie and not fee like I thought it was so it doesn't rhyme :(

Nathan Ramesh - 7 years, 1 month ago

Both pronunciations are correct I think.

Daniel Liu - 7 years, 1 month ago

I think it's pronounced "phi" with an "iy" sound. Rhymes with pi. Actually, I went to a Latin camp and that's what it is. I memorized the whole Greek alphabet. Daniel, what language are you taking?

Finn Hulse - 7 years, 1 month ago

@Finn Hulse – Japanese 1st year. Fluent in English and Chinese (kinda :P)

Daniel Liu - 7 years, 1 month ago

@Daniel Liu – Dang you're lucky to go to such a good school. My school is one of the worst in the country but my French teacher is unbeivably good.

Finn Hulse - 7 years, 1 month ago

Phi with a and b

The answer is 3.

3 solutions

0 pending reports