How do you factor $216 a^{3} b^{3} - 343 c^{6}$ $216a^3b^3-343c^6$ ?

Question

2407 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2018-07-06T11:36:36

$(6 a b - 7 c^{2}) (36 a^{2} b^{2} + 42 a b c^{2} + 49 c^{4})$ $color(blue)((6ab-7c^2)(36a^2b^2+42abc^2+49c^4)$

First notice:

$216 = 6^{3}$ $216=6^3$

and:

$343 = 7^{3}$ $343=7^3$

We can now write:

$6^{3} a^{3} b^{3} - 7^{3} c^{6}$ $6^3a^3b^3-7^3c^6$

Which leads to:

${(6 a b)}^{3} - {(7 c^{2})}^{3}$ $(6ab)^3-(7c^2)^3$

This is the difference of two cubes:

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

$:.$

$(6ab-7c^2)(36a^2b^2+42abc^2+49c^4)$

Answer 2 · 2018-07-06T11:46:30

$(6ab-7c^2)(36a^2b^2+42abc^2+49c^4)$

Note that

$216a^3b^3=(2*3*a*b)^3$
and

$343c^6=(7c^2)^2$

we use the formula

$a^3-b^3=(a-b)(a^2+ab+b^2)$
so we get

$(6ab-7c^2)(36a^2b^2+42abc^2+49c^4)$

How do you factor 216a^3b^3-343c^6216a3b3−343c6216a^3b^3-343c^6?