Solve g(x)=1/x^6-3/4x^3-5

$$g ( x ) = \frac { 1 } { x ^ { 6 } } - \frac { 3 } { 4 x ^ { 3 } } - 5$$

Evaluate

$$-5-\frac{3}{4x^{3}}+\frac{1}{x^{6}}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x^{6} and 4x^{3} is 4x^{6}. Multiply \frac{1}{x^{6}} times \frac{4}{4}. Multiply \frac{3}{4x^{3}} times \frac{x^{3}}{x^{3}}.

$$\frac{4}{4x^{6}}-\frac{3x^{3}}{4x^{6}}-5$$

Since \frac{4}{4x^{6}} and \frac{3x^{3}}{4x^{6}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{4-3x^{3}}{4x^{6}}-5$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 5 times \frac{4x^{6}}{4x^{6}}.

$$\frac{4-3x^{3}}{4x^{6}}-\frac{5\times 4x^{6}}{4x^{6}}$$

Since \frac{4-3x^{3}}{4x^{6}} and \frac{5\times 4x^{6}}{4x^{6}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{4-3x^{3}-5\times 4x^{6}}{4x^{6}}$$

Do the multiplications in 4-3x^{3}-5\times 4x^{6}.

$$\frac{4-3x^{3}-20x^{6}}{4x^{6}}$$

Factor the expressions that are not already factored in \frac{4-3x^{3}-20x^{6}}{4x^{6}}.

$$\frac{-20\left(x^{3}-\left(-\frac{1}{40}\sqrt{329}-\frac{3}{40}\right)\right)\left(x^{3}-\left(\frac{1}{40}\sqrt{329}-\frac{3}{40}\right)\right)}{4x^{6}}$$

Cancel out 4 in both numerator and denominator.

$$\frac{-5\left(x^{3}-\left(-\frac{1}{40}\sqrt{329}-\frac{3}{40}\right)\right)\left(x^{3}-\left(\frac{1}{40}\sqrt{329}-\frac{3}{40}\right)\right)}{x^{6}}$$

To find the opposite of -\frac{1}{40}\sqrt{329}-\frac{3}{40}, find the opposite of each term.

$$\frac{-5\left(x^{3}+\frac{1}{40}\sqrt{329}+\frac{3}{40}\right)\left(x^{3}-\left(\frac{1}{40}\sqrt{329}-\frac{3}{40}\right)\right)}{x^{6}}$$

To find the opposite of \frac{1}{40}\sqrt{329}-\frac{3}{40}, find the opposite of each term.

$$\frac{-5\left(x^{3}+\frac{1}{40}\sqrt{329}+\frac{3}{40}\right)\left(x^{3}-\frac{1}{40}\sqrt{329}+\frac{3}{40}\right)}{x^{6}}$$

Use the distributive property to multiply -5 by x^{3}+\frac{1}{40}\sqrt{329}+\frac{3}{40}.

$$\frac{\left(-5x^{3}-\frac{1}{8}\sqrt{329}-\frac{3}{8}\right)\left(x^{3}-\frac{1}{40}\sqrt{329}+\frac{3}{40}\right)}{x^{6}}$$

Use the distributive property to multiply -5x^{3}-\frac{1}{8}\sqrt{329}-\frac{3}{8} by x^{3}-\frac{1}{40}\sqrt{329}+\frac{3}{40} and combine like terms.

$$\frac{-5x^{6}-\frac{3}{4}x^{3}+\frac{1}{320}\left(\sqrt{329}\right)^{2}-\frac{9}{320}}{x^{6}}$$

The square of \sqrt{329} is 329.

$$\frac{-5x^{6}-\frac{3}{4}x^{3}+\frac{1}{320}\times 329-\frac{9}{320}}{x^{6}}$$

Multiply \frac{1}{320} and 329 to get \frac{329}{320}.

$$\frac{-5x^{6}-\frac{3}{4}x^{3}+\frac{329}{320}-\frac{9}{320}}{x^{6}}$$

Subtract \frac{9}{320} from \frac{329}{320} to get 1.

$$\frac{-5x^{6}-\frac{3}{4}x^{3}+1}{x^{6}}$$

Factor

$$-\frac{5\left(x^{3}-\frac{-\sqrt{329}-3}{40}\right)\left(x^{3}-\frac{\sqrt{329}-3}{40}\right)}{x^{6}}$$