Solve sqrt[3]{frac{8/27}-sqrt{2/3}}{frac{3sqrt{2}-2sqrt{3}}{sqrt{27}}}

$$\frac{ \sqrt[ 3 ]{ \frac{ 8 }{ 27 } } - \sqrt{ \frac{ 2 }{ 3 } } }{ \frac{ 3 \sqrt{ 2 } -2 \sqrt{ 3 } }{ \sqrt{ 27 } } }$$

Evaluate

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Solution Steps

Calculate \sqrt[3]{\frac{8}{27}} and get \frac{2}{3}.

$$\frac{\frac{2}{3}-\sqrt{\frac{2}{3}}}{\frac{3\sqrt{2}-2\sqrt{3}}{\sqrt{27}}}$$

Rewrite the square root of the division \sqrt{\frac{2}{3}} as the division of square roots \frac{\sqrt{2}}{\sqrt{3}}.

$$\frac{\frac{2}{3}-\frac{\sqrt{2}}{\sqrt{3}}}{\frac{3\sqrt{2}-2\sqrt{3}}{\sqrt{27}}}$$

Rationalize the denominator of \frac{\sqrt{2}}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{\frac{2}{3}-\frac{\sqrt{2}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}}{\frac{3\sqrt{2}-2\sqrt{3}}{\sqrt{27}}}$$

The square of \sqrt{3} is 3.

$$\frac{\frac{2}{3}-\frac{\sqrt{2}\sqrt{3}}{3}}{\frac{3\sqrt{2}-2\sqrt{3}}{\sqrt{27}}}$$

To multiply \sqrt{2} and \sqrt{3}, multiply the numbers under the square root.

$$\frac{\frac{2}{3}-\frac{\sqrt{6}}{3}}{\frac{3\sqrt{2}-2\sqrt{3}}{\sqrt{27}}}$$

Since \frac{2}{3} and \frac{\sqrt{6}}{3} have the same denominator, add them by adding their numerators.

$$\frac{\frac{2+\sqrt{6}}{3}}{\frac{3\sqrt{2}-2\sqrt{3}}{\sqrt{27}}}$$

Factor 27=3^{2}\times 3. Rewrite the square root of the product \sqrt{3^{2}\times 3} as the product of square roots \sqrt{3^{2}}\sqrt{3}. Take the square root of 3^{2}.

$$\frac{\frac{2+\sqrt{6}}{3}}{\frac{3\sqrt{2}-2\sqrt{3}}{3\sqrt{3}}}$$

Rationalize the denominator of \frac{3\sqrt{2}-2\sqrt{3}}{3\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{\frac{2+\sqrt{6}}{3}}{\frac{\left(3\sqrt{2}-2\sqrt{3}\right)\sqrt{3}}{3\left(\sqrt{3}\right)^{2}}}$$

The square of \sqrt{3} is 3.

$$\frac{\frac{2+\sqrt{6}}{3}}{\frac{\left(3\sqrt{2}-2\sqrt{3}\right)\sqrt{3}}{3\times 3}}$$

Multiply 3 and 3 to get 9.

$$\frac{\frac{2+\sqrt{6}}{3}}{\frac{\left(3\sqrt{2}-2\sqrt{3}\right)\sqrt{3}}{9}}$$

Divide \frac{2+\sqrt{6}}{3} by \frac{\left(3\sqrt{2}-2\sqrt{3}\right)\sqrt{3}}{9} by multiplying \frac{2+\sqrt{6}}{3} by the reciprocal of \frac{\left(3\sqrt{2}-2\sqrt{3}\right)\sqrt{3}}{9}.

$$\frac{\left(2+\sqrt{6}\right)\times 9}{3\left(3\sqrt{2}-2\sqrt{3}\right)\sqrt{3}}$$

Cancel out 3 in both numerator and denominator.

$$\frac{3\left(\sqrt{6}+2\right)}{\sqrt{3}\left(-2\sqrt{3}+3\sqrt{2}\right)}$$

Rationalize the denominator of \frac{3\left(\sqrt{6}+2\right)}{\sqrt{3}\left(-2\sqrt{3}+3\sqrt{2}\right)} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{3\left(\sqrt{6}+2\right)\sqrt{3}}{\left(\sqrt{3}\right)^{2}\left(-2\sqrt{3}+3\sqrt{2}\right)}$$

The square of \sqrt{3} is 3.

$$\frac{3\left(\sqrt{6}+2\right)\sqrt{3}}{3\left(-2\sqrt{3}+3\sqrt{2}\right)}$$

Use the distributive property to multiply 3 by \sqrt{6}+2.

$$\frac{\left(3\sqrt{6}+6\right)\sqrt{3}}{3\left(-2\sqrt{3}+3\sqrt{2}\right)}$$

Use the distributive property to multiply 3\sqrt{6}+6 by \sqrt{3}.

$$\frac{3\sqrt{6}\sqrt{3}+6\sqrt{3}}{3\left(-2\sqrt{3}+3\sqrt{2}\right)}$$

Factor 6=3\times 2. Rewrite the square root of the product \sqrt{3\times 2} as the product of square roots \sqrt{3}\sqrt{2}.

$$\frac{3\sqrt{3}\sqrt{2}\sqrt{3}+6\sqrt{3}}{3\left(-2\sqrt{3}+3\sqrt{2}\right)}$$

Multiply \sqrt{3} and \sqrt{3} to get 3.

$$\frac{3\times 3\sqrt{2}+6\sqrt{3}}{3\left(-2\sqrt{3}+3\sqrt{2}\right)}$$

Multiply 3 and 3 to get 9.

$$\frac{9\sqrt{2}+6\sqrt{3}}{3\left(-2\sqrt{3}+3\sqrt{2}\right)}$$

Use the distributive property to multiply 3 by -2\sqrt{3}+3\sqrt{2}.

$$\frac{9\sqrt{2}+6\sqrt{3}}{-6\sqrt{3}+9\sqrt{2}}$$

Rationalize the denominator of \frac{9\sqrt{2}+6\sqrt{3}}{-6\sqrt{3}+9\sqrt{2}} by multiplying numerator and denominator by -6\sqrt{3}-9\sqrt{2}.

$$\frac{\left(9\sqrt{2}+6\sqrt{3}\right)\left(-6\sqrt{3}-9\sqrt{2}\right)}{\left(-6\sqrt{3}+9\sqrt{2}\right)\left(-6\sqrt{3}-9\sqrt{2}\right)}$$

Consider \left(-6\sqrt{3}+9\sqrt{2}\right)\left(-6\sqrt{3}-9\sqrt{2}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(9\sqrt{2}+6\sqrt{3}\right)\left(-6\sqrt{3}-9\sqrt{2}\right)}{\left(-6\sqrt{3}\right)^{2}-\left(9\sqrt{2}\right)^{2}}$$

Expand \left(-6\sqrt{3}\right)^{2}.

$$\frac{\left(9\sqrt{2}+6\sqrt{3}\right)\left(-6\sqrt{3}-9\sqrt{2}\right)}{\left(-6\right)^{2}\left(\sqrt{3}\right)^{2}-\left(9\sqrt{2}\right)^{2}}$$

Calculate -6 to the power of 2 and get 36.

$$\frac{\left(9\sqrt{2}+6\sqrt{3}\right)\left(-6\sqrt{3}-9\sqrt{2}\right)}{36\left(\sqrt{3}\right)^{2}-\left(9\sqrt{2}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{\left(9\sqrt{2}+6\sqrt{3}\right)\left(-6\sqrt{3}-9\sqrt{2}\right)}{36\times 3-\left(9\sqrt{2}\right)^{2}}$$

Multiply 36 and 3 to get 108.

$$\frac{\left(9\sqrt{2}+6\sqrt{3}\right)\left(-6\sqrt{3}-9\sqrt{2}\right)}{108-\left(9\sqrt{2}\right)^{2}}$$

Expand \left(9\sqrt{2}\right)^{2}.

$$\frac{\left(9\sqrt{2}+6\sqrt{3}\right)\left(-6\sqrt{3}-9\sqrt{2}\right)}{108-9^{2}\left(\sqrt{2}\right)^{2}}$$

Calculate 9 to the power of 2 and get 81.

$$\frac{\left(9\sqrt{2}+6\sqrt{3}\right)\left(-6\sqrt{3}-9\sqrt{2}\right)}{108-81\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{\left(9\sqrt{2}+6\sqrt{3}\right)\left(-6\sqrt{3}-9\sqrt{2}\right)}{108-81\times 2}$$

Multiply 81 and 2 to get 162.

$$\frac{\left(9\sqrt{2}+6\sqrt{3}\right)\left(-6\sqrt{3}-9\sqrt{2}\right)}{108-162}$$

Subtract 162 from 108 to get -54.

$$\frac{\left(9\sqrt{2}+6\sqrt{3}\right)\left(-6\sqrt{3}-9\sqrt{2}\right)}{-54}$$

Use the distributive property to multiply 9\sqrt{2}+6\sqrt{3} by -6\sqrt{3}-9\sqrt{2} and combine like terms.

$$\frac{-108\sqrt{3}\sqrt{2}-81\left(\sqrt{2}\right)^{2}-36\left(\sqrt{3}\right)^{2}}{-54}$$

To multiply \sqrt{3} and \sqrt{2}, multiply the numbers under the square root.

$$\frac{-108\sqrt{6}-81\left(\sqrt{2}\right)^{2}-36\left(\sqrt{3}\right)^{2}}{-54}$$

The square of \sqrt{2} is 2.

$$\frac{-108\sqrt{6}-81\times 2-36\left(\sqrt{3}\right)^{2}}{-54}$$

Multiply -81 and 2 to get -162.

$$\frac{-108\sqrt{6}-162-36\left(\sqrt{3}\right)^{2}}{-54}$$

The square of \sqrt{3} is 3.

$$\frac{-108\sqrt{6}-162-36\times 3}{-54}$$

Multiply -36 and 3 to get -108.

$$\frac{-108\sqrt{6}-162-108}{-54}$$

Subtract 108 from -162 to get -270.

$$\frac{-108\sqrt{6}-270}{-54}$$

Divide each term of -108\sqrt{6}-270 by -54 to get 2\sqrt{6}+5.

$$2\sqrt{6}+5$$

Factor

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