[tex] \sqrt[3]{ \sqrt[3]{2} - 1} = \sqrt[3]{a} + \sqrt[3]{b} + \sqrt[3]{ \frac{1}{9} } \\ \\ If \: a>b \: \: , \: \: Find \: \: (a+2b) \:.[/tex]

Accepted Solution

Step One
Subtract cube root 1/9 to the left hand side. Or subtract cube root (1/9) from both sides.
[tex] \sqrt[3]{ \sqrt[3]{2} -1 } - \sqrt[3]{ \frac{1}{9} } = \sqrt[3]{a} + \sqrt[3]{b} [/tex]

Step Two. 
There is a minus sign in front of [tex] {-}\sqrt[3]{ \frac{1}{9} } [/tex]
We must get rid of it. Because it is a minus in front of a cube root, we can bring it inside the cube root sign like so, and make it a plus out side the cube root sign
 [tex] {+}\sqrt[3]{ \frac{-1}{9} } [/tex]
Step Three
Write the Left side with the minus sign placed in the proper place
[tex] \sqrt[3]{ \sqrt[3]{2} -1 } + \sqrt[3]{ \frac{-1}{9} } = \sqrt[3]{a} + \sqrt[3]{b} [/tex]

Step Four
Equate cube root b with cube root (-1/9)
[tex] \sqrt[3]{b} = \sqrt[3]{ \frac{-1}{9} } [/tex]

Step Five
Equate the cube root of a with what's left over on the left
[tex] \sqrt[3]{ \sqrt[3]{2} -1 } = \sqrt[3]{a} [/tex]

Step 6. 
I'll just work with b for a moment.
Cube both sides of  cube root (b) = cube root (-1/9)
[tex] \sqrt[3]{b} ^{3} =\sqrt[3]{ \frac{-1}{9} }^3} [/tex]
[tex] \text{b =} \frac{-1}{9}[/tex]
[tex] \text{2b =}\frac{-2}{9}[/tex]

Step seven
the other part is done exactly the same way
a = cuberoot(2) - 1.

What you do from here is up to you. It is not pleasant.
Is this clearer?

a + 2b should come to cuberoot(2) - 1 - 2/9
a + 2b should come to cuberoot(2) - 11/9

I hope a person is marking this. I wonder how many of your class mates got it.