Solve this problem using the difference of cubes formula, since both terms are perfect cubes.
According to the difference of cubes formula, given a³ - b³ = (a - b)(a² + ab+ b²).
So, follow the steps to solve this problem.
Step 01: Find "a".
Comparing with the equation above:
[tex]a^3=27x^3[/tex]Take the cubic root from both sides:
[tex]\begin{gathered} \sqrt[3]{a^3}=\sqrt[3]{27x^3} \\ a^{\frac{3}{3}=}\sqrt[3]{27}\cdot\sqrt[3]{x^3} \end{gathered}[/tex]Factoring 27 = 3 * 3 *3 = 3³. Then,
[tex]\begin{gathered} a=\sqrt[3]{3^3}\cdot\sqrt[3]{x^3} \\ a=3x \end{gathered}[/tex]Step 01: Find "b".
[tex]b^3=64[/tex]Taking the cubic root from both sides and factoring 64:
64 = 4*4*4. Then,
[tex]\begin{gathered} \sqrt[3]{b^3}=\sqrt[3]{4^3} \\ b=4 \end{gathered}[/tex]Step 03: Substitute "a" and "b" in the formula.
a³ - b³ = (a - b)(a² + ab+ b²).
[tex]\begin{gathered} 27x^3-64=(3x-4)\cdot\lbrack(3x)^2+3x\cdot4+4^2\rbrack \\ =(3x-4)\cdot(9x^2+12x+16) \end{gathered}[/tex]Answer:
[tex]27x^3-64=(3x-4)\cdot(9x^2+12x+16)[/tex]