Solve (2sqrt{24}-3sqrt{27})divsqrt{6}

$$( 2 \sqrt { 24 } - 3 \sqrt { 27 } ) \div \sqrt { 6 }$$

Evaluate

$$-\frac{9\sqrt{2}}{2}+4\approx -2.363961031$$

Solution Steps

Factor 24=2^{2}\times 6. Rewrite the square root of the product \sqrt{2^{2}\times 6} as the product of square roots \sqrt{2^{2}}\sqrt{6}. Take the square root of 2^{2}.

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

Multiply 2 and 2 to get 4.

$$\frac{4\sqrt{6}-3\sqrt{27}}{\sqrt{6}}$$

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}.

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

Multiply -3 and 3 to get -9.

$$\frac{4\sqrt{6}-9\sqrt{3}}{\sqrt{6}}$$

Rationalize the denominator of \frac{4\sqrt{6}-9\sqrt{3}}{\sqrt{6}} by multiplying numerator and denominator by \sqrt{6}.

$$\frac{\left(4\sqrt{6}-9\sqrt{3}\right)\sqrt{6}}{\left(\sqrt{6}\right)^{2}}$$

The square of \sqrt{6} is 6.

$$\frac{\left(4\sqrt{6}-9\sqrt{3}\right)\sqrt{6}}{6}$$

Use the distributive property to multiply 4\sqrt{6}-9\sqrt{3} by \sqrt{6}.

$$\frac{4\left(\sqrt{6}\right)^{2}-9\sqrt{3}\sqrt{6}}{6}$$

The square of \sqrt{6} is 6.

$$\frac{4\times 6-9\sqrt{3}\sqrt{6}}{6}$$

Multiply 4 and 6 to get 24.

$$\frac{24-9\sqrt{3}\sqrt{6}}{6}$$

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}.

$$\frac{24-9\sqrt{3}\sqrt{3}\sqrt{2}}{6}$$

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

$$\frac{24-9\times 3\sqrt{2}}{6}$$

Multiply -9 and 3 to get -27.

$$\frac{24-27\sqrt{2}}{6}$$