Note the following cube root calculations The cube root of the first term x^{3} = x ← This is {a} The cube root of the second term 27y^{9} = 3y^{3} ← This is {b}

Since both cube roots are integer constants and powers, x^{3} - 27y^{9} is in the Difference of cubes format

The formula for factoring the Difference of cubes is as follows: a^{3} - b^{3} = (a - b)(a^{2} + ab + b^{2})

Calculate ab ab = (x)(3y^{3}) ab = (1 x 3)xy^{3} ab = 3xy^{3}

Calculate the square of the a term: The square of the a term = (x)^{2} = x^{(1 x 2)} The square of the a term = (x)^{2} = x^{2}

Calculate the square of the b term: The square of the b term = (3y^{3})^{2} = 3^{2}y^{(3 x 2)} The square of the b term = (3y^{3})^{2} = 9y^{6}

Our factored expression using the Difference of cubes formula becomes: (x - 3y^{3})(x^{2} + 3xy^{3} + 9y^{6}) Our factored out term becomes: (x - 3y^{3})(x^{2} + 3xy^{3} + 9y^{6})