Solve (2sqrt{54}-3sqrt{18})divsqrt{6}

$$( 2 \sqrt { 54 } - 3 \sqrt { 18 } ) \div \sqrt { 6 }$$

Evaluate

$$6-3\sqrt{3}\approx 0.803847577$$

Solution Steps

Factor 54=3^{2}\times 6. Rewrite the square root of the product \sqrt{3^{2}\times 6} as the product of square roots \sqrt{3^{2}}\sqrt{6}. Take the square root of 3^{2}.

$$\frac{2\times 3\sqrt{6}-3\sqrt{18}}{\sqrt{6}}$$

Multiply 2 and 3 to get 6.

$$\frac{6\sqrt{6}-3\sqrt{18}}{\sqrt{6}}$$

Factor 18=3^{2}\times 2. Rewrite the square root of the product \sqrt{3^{2}\times 2} as the product of square roots \sqrt{3^{2}}\sqrt{2}. Take the square root of 3^{2}.

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

Multiply -3 and 3 to get -9.

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

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

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

The square of \sqrt{6} is 6.

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

Use the distributive property to multiply 6\sqrt{6}-9\sqrt{2} by \sqrt{6}.

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

The square of \sqrt{6} is 6.

$$\frac{6\times 6-9\sqrt{2}\sqrt{6}}{6}$$

Multiply 6 and 6 to get 36.

$$\frac{36-9\sqrt{2}\sqrt{6}}{6}$$

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}.

$$\frac{36-9\sqrt{2}\sqrt{2}\sqrt{3}}{6}$$

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

$$\frac{36-9\times 2\sqrt{3}}{6}$$

Multiply -9 and 2 to get -18.

$$\frac{36-18\sqrt{3}}{6}$$