Simplify Fractions: 2/(49^1/3+7^1/3+1)

[songoku]

Homework Statement

Simplify

$$\frac{2}{49^{1/3}+7^{1/3}+1}$$

Homework Equations

The Attempt at a Solution

I can simplify
$$\frac{2}{49^{1/3}+7^{1/3}}$$

but I don't have idea how to do this one..

EDIT: I have found it. No need to reply this. Thanks :)

 

[SammyS]

Homework Statement

Simplify

$$\frac{2}{49^{1/3}+7^{1/3}+1}$$
...

The Attempt at a Solution

I can simplify
$$\frac{2}{49^{1/3}+7^{1/3}}$$

but I don't have idea how to do this one..

EDIT: I have found it. No need to reply this. Thanks :)

I see that you have found it, but for others looking later:

Multiply the numerator & denominator by $\displaystyle (7^{1/3}-1)$

 

[songoku]

I see that you have found it, but for others looking later:

Multiply the numerator & denominator by $\displaystyle (7^{1/3}-1)$

I did it another way but let discuss your way. I drop the numerator because we only concern about the denominator.

After multiplying it with 7^1/3 - 1, I get:
49^1/3 . 7^1/3 - 49^1/3 + 7^2/3 - 1

I think I have to multiply the denominator with another term but I can't find it..

 

[SammyS]

49=7² so the denominator becomes 6 .

49^1/3 ∙ 7^1/3 - 49^1/3 + 7^2/3 - 1

= 7^2/3 ∙ 7^1/3 - 7^2/3 + 7^2/3 - 1

= 7 - 1

 

[songoku]

49=7² so the denominator becomes 6 .

49^1/3 ∙ 7^1/3 - 49^1/3 + 7^2/3 - 1

= 7^2/3 ∙ 7^1/3 - 7^2/3 + 7^2/3 - 1

= 7 - 1

It really didn't cross my mind; multiplying with (7^1/3 - 1) will directly give the answer. Your method is much simpler than mine. I should have noticed the pattern a^2 + a + 1 in the denominator, which can be simplified by multiplying it with (a - 1)

Thanks :)