Solve (a^3/sqrt{3})^3-a^2*a/sqrt{3}

$$( \frac { a ^ { 3 } } { \sqrt { 3 } } ) ^ { 3 } - a ^ { 2 } \cdot \frac { a } { \sqrt { 3 } }$$

Factor

$$\frac{\sqrt{3}a^{3}\left(a^{6}-3\right)}{9}$$

Solution Steps

Rationalize the denominator of \frac{a^{3}}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$factor(\left(\frac{a^{3}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}\right)^{3}-a^{2}\times \frac{a}{\sqrt{3}})$$

The square of \sqrt{3} is 3.

$$factor(\left(\frac{a^{3}\sqrt{3}}{3}\right)^{3}-a^{2}\times \frac{a}{\sqrt{3}})$$

To raise \frac{a^{3}\sqrt{3}}{3} to a power, raise both numerator and denominator to the power and then divide.

$$factor(\frac{\left(a^{3}\sqrt{3}\right)^{3}}{3^{3}}-a^{2}\times \frac{a}{\sqrt{3}})$$

Rationalize the denominator of \frac{a}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$factor(\frac{\left(a^{3}\sqrt{3}\right)^{3}}{3^{3}}-a^{2}\times \frac{a\sqrt{3}}{\left(\sqrt{3}\right)^{2}})$$

The square of \sqrt{3} is 3.

$$factor(\frac{\left(a^{3}\sqrt{3}\right)^{3}}{3^{3}}-a^{2}\times \frac{a\sqrt{3}}{3})$$

Express a^{2}\times \frac{a\sqrt{3}}{3} as a single fraction.

$$factor(\frac{\left(a^{3}\sqrt{3}\right)^{3}}{3^{3}}-\frac{a^{2}a\sqrt{3}}{3})$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$factor(\frac{\left(a^{3}\sqrt{3}\right)^{3}}{3^{3}}-\frac{a^{3}\sqrt{3}}{3})$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3^{3} and 3 is 27. Multiply \frac{a^{3}\sqrt{3}}{3} times \frac{9}{9}.

$$factor(\frac{\left(a^{3}\sqrt{3}\right)^{3}}{27}-\frac{9a^{3}\sqrt{3}}{27})$$

Since \frac{\left(a^{3}\sqrt{3}\right)^{3}}{27} and \frac{9a^{3}\sqrt{3}}{27} have the same denominator, subtract them by subtracting their numerators.

$$factor(\frac{\left(a^{3}\sqrt{3}\right)^{3}-9a^{3}\sqrt{3}}{27})$$

Expand \left(a^{3}\sqrt{3}\right)^{3}.

$$factor(\frac{\left(a^{3}\right)^{3}\left(\sqrt{3}\right)^{3}-9a^{3}\sqrt{3}}{27})$$

To raise a power to another power, multiply the exponents. Multiply 3 and 3 to get 9.

$$factor(\frac{a^{9}\left(\sqrt{3}\right)^{3}-9a^{3}\sqrt{3}}{27})$$

Consider a^{9}\left(\sqrt{3}\right)^{3}-9a^{3}\sqrt{3}. Factor out a^{3}.

$$a^{3}\left(a^{6}\left(\sqrt{3}\right)^{3}-9\sqrt{3}\right)$$

Consider a^{6}\times 3^{\frac{3}{2}}-9\times 3^{\frac{1}{2}}. Factor out 3.

$$3\left(a^{6}\sqrt{3}-3\sqrt{3}\right)$$

Consider a^{6}\sqrt{3}-3\sqrt{3}. Factor out \sqrt{3}.

$$\sqrt{3}\left(a^{6}-3\right)$$

Rewrite the complete factored expression. Simplify. Polynomial a^{6}-3 is not factored since it does not have any rational roots.

$$\frac{\left(a^{6}-3\right)\sqrt{3}a^{3}}{9}$$

Evaluate

$$\frac{\sqrt{3}a^{3}\left(a^{6}-3\right)}{9}$$