Solve 3*(1/sqrt{2})^2+sqrt{27/9}divsqrt[3]{-8}=?

$$3 \times ( \frac { 1 } { \sqrt { 2 } } ) ^ { 2 } + \sqrt { 2 \frac { 7 } { 9 } } \div \sqrt[ 3 ] { - 8 } = ?$$

Evaluate

$$\frac{2}{3}\approx 0.666666667$$

Solution Steps

Rationalize the denominator of \frac{1}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$3\times \left(\frac{\sqrt{2}}{\left(\sqrt{2}\right)^{2}}\right)^{2}+\frac{\sqrt{\frac{2\times 9+7}{9}}}{\sqrt[3]{-8}}$$

The square of \sqrt{2} is 2.

$$3\times \left(\frac{\sqrt{2}}{2}\right)^{2}+\frac{\sqrt{\frac{2\times 9+7}{9}}}{\sqrt[3]{-8}}$$

To raise \frac{\sqrt{2}}{2} to a power, raise both numerator and denominator to the power and then divide.

$$3\times \frac{\left(\sqrt{2}\right)^{2}}{2^{2}}+\frac{\sqrt{\frac{2\times 9+7}{9}}}{\sqrt[3]{-8}}$$

Express 3\times \frac{\left(\sqrt{2}\right)^{2}}{2^{2}} as a single fraction.

$$\frac{3\left(\sqrt{2}\right)^{2}}{2^{2}}+\frac{\sqrt{\frac{2\times 9+7}{9}}}{\sqrt[3]{-8}}$$

Multiply 2 and 9 to get 18.

$$\frac{3\left(\sqrt{2}\right)^{2}}{2^{2}}+\frac{\sqrt{\frac{18+7}{9}}}{\sqrt[3]{-8}}$$

Add 18 and 7 to get 25.

$$\frac{3\left(\sqrt{2}\right)^{2}}{2^{2}}+\frac{\sqrt{\frac{25}{9}}}{\sqrt[3]{-8}}$$

Rewrite the square root of the division \frac{25}{9} as the division of square roots \frac{\sqrt{25}}{\sqrt{9}}. Take the square root of both numerator and denominator.

$$\frac{3\left(\sqrt{2}\right)^{2}}{2^{2}}+\frac{\frac{5}{3}}{\sqrt[3]{-8}}$$

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

$$\frac{3\left(\sqrt{2}\right)^{2}}{2^{2}}+\frac{\frac{5}{3}}{-2}$$

Express \frac{\frac{5}{3}}{-2} as a single fraction.

$$\frac{3\left(\sqrt{2}\right)^{2}}{2^{2}}+\frac{5}{3\left(-2\right)}$$

Multiply 3 and -2 to get -6.

$$\frac{3\left(\sqrt{2}\right)^{2}}{2^{2}}+\frac{5}{-6}$$

Fraction \frac{5}{-6} can be rewritten as -\frac{5}{6} by extracting the negative sign.

$$\frac{3\left(\sqrt{2}\right)^{2}}{2^{2}}-\frac{5}{6}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2^{2} and 6 is 12. Multiply \frac{3\left(\sqrt{2}\right)^{2}}{2^{2}} times \frac{3}{3}. Multiply \frac{5}{6} times \frac{2}{2}.

$$\frac{3\times 3\left(\sqrt{2}\right)^{2}}{12}-\frac{5\times 2}{12}$$

Since \frac{3\times 3\left(\sqrt{2}\right)^{2}}{12} and \frac{5\times 2}{12} have the same denominator, subtract them by subtracting their numerators.

$$\frac{3\times 3\left(\sqrt{2}\right)^{2}-5\times 2}{12}$$

The square of \sqrt{2} is 2.

$$\frac{3\times 2}{2^{2}}-\frac{5}{6}$$

Multiply 3 and 2 to get 6.

$$\frac{6}{2^{2}}-\frac{5}{6}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{6}{4}-\frac{5}{6}$$

Reduce the fraction \frac{6}{4} to lowest terms by extracting and canceling out 2.

$$\frac{3}{2}-\frac{5}{6}$$

Subtract \frac{5}{6} from \frac{3}{2} to get \frac{2}{3}.

$$\frac{2}{3}$$

Factor

$$\frac{2}{3} = 0.6666666666666666$$