Step 1 :

Factor out common factor $ \color{blue}{ a^2 } $:

$$ a^8-a^2b^6 = a^2 ( a^6-b^6 ) $$

Step 2 :

Rewrite $ a^6-b^6 $ as:

$$ \color{blue}{ a^6-b^6 = (a^3)^2 - (b^3)^2 } $$

Now we can apply the difference of squares formula.

$$ I^2 - II^2 = (I - II)(I + II) $$

After putting $ I = a^3 $ and $ II = b^3 $ , we have:

$$ a^6-b^6 = (a^3)^2 - (b^3)^2 = ( a^3-b^3 ) ( a^3+b^3 ) $$

Step 3 :

To factor $ a^{3}+b^{3} $ we can use sum of cubes formula:

$$ I^3 + II^3 = (I + II) (I^2 - I \cdot II + II^2)$$

After putting $ I = a $ and $ II = b $ , we have:

$$ a^{3}+b^{3} = ( a+b ) ( a^{2}-ab+b^{2} ) $$

Step 4 :

To factor $ a^{3}-b^{3} $ we can use difference of cubes formula:

$$ I^3 - II^3 = (I - II) (I^2 + I \cdot II + II^2) $$

After putting $ I = a $ and $ II = b $ , we have:

$$ a^{3}-b^{3} = ( a-b ) ( a^{2}+ab+b^{2} ) $$

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