Solve (sqrt{27}-1/3sqrt{6})divsqrt{8}

$$( \sqrt { 27 } - \frac { 1 } { 3 } \sqrt { 6 } ) \div \sqrt { 8 }$$

Evaluate

$$\frac{3\sqrt{6}}{4}-\frac{\sqrt{3}}{6}\approx 1.548442172$$

Solution Steps

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}.

$$\frac{3\sqrt{3}-\frac{1}{3}\sqrt{6}}{\sqrt{8}}$$

Factor 8=2^{2}\times 2. Rewrite the square root of the product \sqrt{2^{2}\times 2} as the product of square roots \sqrt{2^{2}}\sqrt{2}. Take the square root of 2^{2}.

$$\frac{3\sqrt{3}-\frac{1}{3}\sqrt{6}}{2\sqrt{2}}$$

Rationalize the denominator of \frac{3\sqrt{3}-\frac{1}{3}\sqrt{6}}{2\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{\left(3\sqrt{3}-\frac{1}{3}\sqrt{6}\right)\sqrt{2}}{2\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{\left(3\sqrt{3}-\frac{1}{3}\sqrt{6}\right)\sqrt{2}}{2\times 2}$$

Multiply 2 and 2 to get 4.

$$\frac{\left(3\sqrt{3}-\frac{1}{3}\sqrt{6}\right)\sqrt{2}}{4}$$

Use the distributive property to multiply 3\sqrt{3}-\frac{1}{3}\sqrt{6} by \sqrt{2}.

$$\frac{3\sqrt{3}\sqrt{2}-\frac{1}{3}\sqrt{6}\sqrt{2}}{4}$$

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

$$\frac{3\sqrt{6}-\frac{1}{3}\sqrt{6}\sqrt{2}}{4}$$

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

$$\frac{3\sqrt{6}-\frac{1}{3}\sqrt{2}\sqrt{3}\sqrt{2}}{4}$$

Multiply \sqrt{2} and \sqrt{2} to get 2.

$$\frac{3\sqrt{6}-\frac{1}{3}\times 2\sqrt{3}}{4}$$

Express -\frac{1}{3}\times 2 as a single fraction.

$$\frac{3\sqrt{6}+\frac{-2}{3}\sqrt{3}}{4}$$

Fraction \frac{-2}{3} can be rewritten as -\frac{2}{3} by extracting the negative sign.

$$\frac{3\sqrt{6}-\frac{2}{3}\sqrt{3}}{4}$$

Factor

$$\frac{9 \sqrt{6} - 2 \sqrt{3}}{12} = 1.5484421724925708$$