Solve 4n^2/3-8n^3/9+16n^7/15-2n^5/3

$$\frac { 4 n ^ { 2 } } { 3 } - \frac { 8 n ^ { 3 } } { 9 } + \frac { 16 n ^ { 7 } } { 15 } - \frac { 2 n ^ { 5 } } { 3 }$$

Evaluate

$$\frac{16n^{7}}{15}-\frac{2n^{5}}{3}-\frac{8n^{3}}{9}+\frac{4n^{2}}{3}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3 and 9 is 9. Multiply \frac{4n^{2}}{3} times \frac{3}{3}.

$$\frac{3\times 4n^{2}}{9}-\frac{8n^{3}}{9}+\frac{16n^{7}}{15}-\frac{2n^{5}}{3}$$

Since \frac{3\times 4n^{2}}{9} and \frac{8n^{3}}{9} have the same denominator, subtract them by subtracting their numerators.

$$\frac{3\times 4n^{2}-8n^{3}}{9}+\frac{16n^{7}}{15}-\frac{2n^{5}}{3}$$

Do the multiplications in 3\times 4n^{2}-8n^{3}.

$$\frac{12n^{2}-8n^{3}}{9}+\frac{16n^{7}}{15}-\frac{2n^{5}}{3}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 9 and 15 is 45. Multiply \frac{12n^{2}-8n^{3}}{9} times \frac{5}{5}. Multiply \frac{16n^{7}}{15} times \frac{3}{3}.

$$\frac{5\left(12n^{2}-8n^{3}\right)}{45}+\frac{3\times 16n^{7}}{45}-\frac{2n^{5}}{3}$$

Since \frac{5\left(12n^{2}-8n^{3}\right)}{45} and \frac{3\times 16n^{7}}{45} have the same denominator, add them by adding their numerators.

$$\frac{5\left(12n^{2}-8n^{3}\right)+3\times 16n^{7}}{45}-\frac{2n^{5}}{3}$$

Do the multiplications in 5\left(12n^{2}-8n^{3}\right)+3\times 16n^{7}.

$$\frac{60n^{2}-40n^{3}+48n^{7}}{45}-\frac{2n^{5}}{3}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 45 and 3 is 45. Multiply \frac{2n^{5}}{3} times \frac{15}{15}.

$$\frac{60n^{2}-40n^{3}+48n^{7}}{45}-\frac{15\times 2n^{5}}{45}$$

Since \frac{60n^{2}-40n^{3}+48n^{7}}{45} and \frac{15\times 2n^{5}}{45} have the same denominator, subtract them by subtracting their numerators.

$$\frac{60n^{2}-40n^{3}+48n^{7}-15\times 2n^{5}}{45}$$

Do the multiplications in 60n^{2}-40n^{3}+48n^{7}-15\times 2n^{5}.

$$\frac{60n^{2}-40n^{3}+48n^{7}-30n^{5}}{45}$$

Factor

$$\frac{2n^{2}\left(24n^{5}-15n^{3}-20n+30\right)}{45}$$

Solution Steps

Factor out \frac{2}{45}.

$$\frac{2\left(30n^{2}-20n^{3}+24n^{7}-15n^{5}\right)}{45}$$

Consider 30n^{2}-20n^{3}+24n^{7}-15n^{5}. Factor out n^{2}.

$$n^{2}\left(30-20n+24n^{5}-15n^{3}\right)$$

Rewrite the complete factored expression. Polynomial 30-20n+24n^{5}-15n^{3} is not factored since it does not have any rational roots.

$$\frac{2n^{2}\left(30-20n+24n^{5}-15n^{3}\right)}{45}$$