Solve (3sqrt{27}-2sqrt{18})divsqrt{3}

$$( 3 \sqrt { 27 } - 2 \sqrt { 18 } ) \div \sqrt { 3 }$$

Evaluate

$$9-2\sqrt{6}\approx 4.101020514$$

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}.

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

Multiply 3 and 3 to get 9.

$$\frac{9\sqrt{3}-2\sqrt{18}}{\sqrt{3}}$$

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}.

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

Multiply -2 and 3 to get -6.

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

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

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

The square of \sqrt{3} is 3.

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

Use the distributive property to multiply 9\sqrt{3}-6\sqrt{2} by \sqrt{3}.

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

The square of \sqrt{3} is 3.

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

Multiply 9 and 3 to get 27.

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

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

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

Divide each term of 27-6\sqrt{6} by 3 to get 9-2\sqrt{6}.

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