Solve (3sqrt{6}-8sqrt{3})divsqrt{24}

$$(3 \sqrt{ 6 } -8 \sqrt{ 3 } ) \div \sqrt{ 24 }$$

Evaluate

$$\frac{3}{2}-2\sqrt{2}\approx -1.328427125$$

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}.

$$\frac{3\sqrt{6}-8\sqrt{3}}{2\sqrt{6}}$$

Rationalize the denominator of \frac{3\sqrt{6}-8\sqrt{3}}{2\sqrt{6}} by multiplying numerator and denominator by \sqrt{6}.

$$\frac{\left(3\sqrt{6}-8\sqrt{3}\right)\sqrt{6}}{2\left(\sqrt{6}\right)^{2}}$$

The square of \sqrt{6} is 6.

$$\frac{\left(3\sqrt{6}-8\sqrt{3}\right)\sqrt{6}}{2\times 6}$$

Multiply 2 and 6 to get 12.

$$\frac{\left(3\sqrt{6}-8\sqrt{3}\right)\sqrt{6}}{12}$$

Use the distributive property to multiply 3\sqrt{6}-8\sqrt{3} by \sqrt{6}.

$$\frac{3\left(\sqrt{6}\right)^{2}-8\sqrt{3}\sqrt{6}}{12}$$

The square of \sqrt{6} is 6.

$$\frac{3\times 6-8\sqrt{3}\sqrt{6}}{12}$$

Multiply 3 and 6 to get 18.

$$\frac{18-8\sqrt{3}\sqrt{6}}{12}$$

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}.

$$\frac{18-8\sqrt{3}\sqrt{3}\sqrt{2}}{12}$$

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

$$\frac{18-8\times 3\sqrt{2}}{12}$$

Multiply -8 and 3 to get -24.

$$\frac{18-24\sqrt{2}}{12}$$

Factor

$$\frac{3 - 4 \sqrt{2}}{2} = -1.3284271247461903$$