Question

Facotr. 1/512 x^9 - 512

Answer 1

so (1/512)x^9 - 512?

Answer 2

yes

Answer 3

do you know the formula for the difference of perfect cubes?

Answer 4

yes

Answer 5

\(\bf \cfrac{1}{512}x^9-512\qquad \textit{keep in mind that}\quad 512 = 8^3,\quad x^9 = (x^3)^3\\ \quad \\
\textit{also recall that }\qquad a^3+b^3 = (a+b)(a^2-ab+b^2)\qquad thus\\ \quad \\
\cfrac{1}{512}x^9-512\implies \cfrac{1}{8^3}(x^3)^3-8^3\implies \left(\cfrac{1}{8}x^3\right)^3-8^3\)

Answer 6

well \(\bf (a-b)(a^2+ab+b^2)= a^3-b^3\) rather

Answer 7

You can reduce both terms to perfect cubes as follows:

(1/512) = (1/8)^3
x^9 = (x^3)^3
512 = 8^3

so

(1/512)x^9 - 512 =

\[\left( \frac{ 1 }{ 8 } \right)^{3} * (x^{3})^{3} - 8^3 = (\frac{ 1 }{ 8 } x^{3})^{3} - 8^3\]

Answer 8

but the answer is, (1/2 x - 2 ) (1/4x^2+x+4)(1/64x^6+x^3+64)

Answer 9

\(\bf \cfrac{1}{512}x^9-512\qquad \textit{keep in mind that}\quad 512 = 8^3,\quad x^9 = (x^3)^3\\ \quad \\
\textit{also recall that }\qquad a^3+b^3 = (a-b)(a^2+ab+b^2)\qquad thus\\ \quad \\
\cfrac{1}{512}x^9-512\implies \cfrac{1}{8^3}(x^3)^3-8^3\implies \left(\cfrac{1}{8}x^3\right)^3-8^3\\ \quad \\

\left(\cfrac{x^3}{8}\right)^3-8^3\implies \left(\cfrac{x^3}{8}-8\right)

\color{blue}{\left(\left(\cfrac{x^3}{8}\right)^2+x^3+8^2\right)}\\ \quad \\
\textit{keep in mind that }\left(\cfrac{x^3}{8}-8\right)\implies \left(\cfrac{x^3}{2^3}-2^3\right)\implies \left(\cfrac{x}{2}\right)^3-2^3\\ \quad \\
\left(\cfrac{x}{2}\right)^3-2^3\implies

\color{blue}{ \left(\cfrac{x}{2}-2\right)\left(\left(\cfrac{x}{2}\right)^2+x+2^2\right)}\)

Answer 10

then expand thatt? how did they get (1/2 x - 2 ) (1/4x^2+x+4)(1/64x^6+x^3+64)

Answer 11

you were asked to factor, so follow jdoe0001's example. his final result is identical to the answer you provided, which was (1/2 x - 2 ) (1/4x^2+x+4)