Factor 8x^24-27y^6

$8 x^{24} - 27 y^{6}$

Step 1

Rewrite $8 x^{24}$ as ${(2 x^{8})}^{3}$ .

${(2 x^{8})}^{3} - 27 y^{6}$

Step 2

Rewrite $27 y^{6}$ as ${(3 y^{2})}^{3}$ .

${(2 x^{8})}^{3} - {(3 y^{2})}^{3}$

Step 3

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = 2 x^{8}$ and $b = 3 y^{2}$ .

$(2 x^{8} - (3 y^{2})) ({(2 x^{8})}^{2} + 2 x^{8} (3 y^{2}) + {(3 y^{2})}^{2})$

Step 4

Simplify.

Multiply $3$ by $- 1$ .

$(2 x^{8} - 3 y^{2}) ({(2 x^{8})}^{2} + 2 x^{8} (3 y^{2}) + {(3 y^{2})}^{2})$

Apply the product rule to $2 x^{8}$ .

$(2 x^{8} - 3 y^{2}) (2^{2} {(x^{8})}^{2} + 2 x^{8} (3 y^{2}) + {(3 y^{2})}^{2})$

Raise $2$ to the power of $2$ .

$(2 x^{8} - 3 y^{2}) (4 {(x^{8})}^{2} + 2 x^{8} (3 y^{2}) + {(3 y^{2})}^{2})$

Multiply the exponents in ${(x^{8})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$(2 x^{8} - 3 y^{2}) (4 x^{8 \cdot 2} + 2 x^{8} (3 y^{2}) + {(3 y^{2})}^{2})$

Multiply $8$ by $2$ .

$(2 x^{8} - 3 y^{2}) (4 x^{16} + 2 x^{8} (3 y^{2}) + {(3 y^{2})}^{2})$

Rewrite using the commutative property of multiplication.

$(2 x^{8} - 3 y^{2}) (4 x^{16} + 2 \cdot 3 x^{8} y^{2} + {(3 y^{2})}^{2})$

Multiply $2$ by $3$ .

$(2 x^{8} - 3 y^{2}) (4 x^{16} + 6 x^{8} y^{2} + {(3 y^{2})}^{2})$

Apply the product rule to $3 y^{2}$ .

$(2 x^{8} - 3 y^{2}) (4 x^{16} + 6 x^{8} y^{2} + 3^{2} {(y^{2})}^{2})$

Raise $3$ to the power of $2$ .

$(2 x^{8} - 3 y^{2}) (4 x^{16} + 6 x^{8} y^{2} + 9 {(y^{2})}^{2})$

Multiply the exponents in ${(y^{2})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$(2 x^{8} - 3 y^{2}) (4 x^{16} + 6 x^{8} y^{2} + 9 y^{2 \cdot 2})$

Multiply $2$ by $2$ .

$(2 x^{8} - 3 y^{2}) (4 x^{16} + 6 x^{8} y^{2} + 9 y^{4})$

Factor 8x^24-27y^6

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