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.