Solve 3/125x^8-24/27x^5

$$\frac{ 3 }{ 125 } { x }^{ 8 } - \frac{ 24 }{ 27 } { x }^{ 5 }$$

Factor

$$\frac{\left(3x-10\right)\left(9x^{2}+30x+100\right)x^{5}}{1125}$$

Solution Steps

Factor out \frac{1}{1125}.

$$\frac{27x^{8}-1000x^{5}}{1125}$$

Consider 27x^{8}-1000x^{5}. Factor out x^{5}.

$$x^{5}\left(27x^{3}-1000\right)$$

Consider 27x^{3}-1000. Rewrite 27x^{3}-1000 as \left(3x\right)^{3}-10^{3}. The difference of cubes can be factored using the rule: a^{3}-b^{3}=\left(a-b\right)\left(a^{2}+ab+b^{2}\right).

$$\left(3x-10\right)\left(9x^{2}+30x+100\right)$$

Rewrite the complete factored expression. Polynomial 9x^{2}+30x+100 is not factored since it does not have any rational roots.

$$\frac{x^{5}\left(3x-10\right)\left(9x^{2}+30x+100\right)}{1125}$$

Evaluate

$$\frac{3x^{8}}{125}-\frac{8x^{5}}{9}$$

Solution Steps

Reduce the fraction \frac{24}{27} to lowest terms by extracting and canceling out 3.

$$\frac{3}{125}x^{8}-\frac{8}{9}x^{5}$$