Solve (sqrt{48+1/4}sqrt{6})divsqrt{27}

$$( \sqrt{ 48+ \frac{ 1 }{ 4 } } \sqrt{ 6 } ) \div \sqrt{ 27 }$$

Evaluate

$$\frac{\sqrt{386}}{6}\approx 3.274480451$$

Solution Steps

Convert 48 to fraction \frac{192}{4}.

$$\frac{\sqrt{\frac{192}{4}+\frac{1}{4}}\sqrt{6}}{\sqrt{27}}$$

Since \frac{192}{4} and \frac{1}{4} have the same denominator, add them by adding their numerators.

$$\frac{\sqrt{\frac{192+1}{4}}\sqrt{6}}{\sqrt{27}}$$

Add 192 and 1 to get 193.

$$\frac{\sqrt{\frac{193}{4}}\sqrt{6}}{\sqrt{27}}$$

Rewrite the square root of the division \sqrt{\frac{193}{4}} as the division of square roots \frac{\sqrt{193}}{\sqrt{4}}.

$$\frac{\frac{\sqrt{193}}{\sqrt{4}}\sqrt{6}}{\sqrt{27}}$$

Calculate the square root of 4 and get 2.

$$\frac{\frac{\sqrt{193}}{2}\sqrt{6}}{\sqrt{27}}$$

Express \frac{\sqrt{193}}{2}\sqrt{6} as a single fraction.

$$\frac{\frac{\sqrt{193}\sqrt{6}}{2}}{\sqrt{27}}$$

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}.

$$\frac{\frac{\sqrt{193}\sqrt{6}}{2}}{3\sqrt{3}}$$

Express \frac{\frac{\sqrt{193}\sqrt{6}}{2}}{3\sqrt{3}} as a single fraction.

$$\frac{\sqrt{193}\sqrt{6}}{2\times 3\sqrt{3}}$$

Rationalize the denominator of \frac{\sqrt{193}\sqrt{6}}{2\times 3\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{\sqrt{193}\sqrt{6}\sqrt{3}}{2\times 3\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{\sqrt{193}\sqrt{6}\sqrt{3}}{2\times 3\times 3}$$

To multiply \sqrt{193} and \sqrt{6}, multiply the numbers under the square root.

$$\frac{\sqrt{1158}\sqrt{3}}{2\times 3\times 3}$$

Factor 1158=3\times 386. Rewrite the square root of the product \sqrt{3\times 386} as the product of square roots \sqrt{3}\sqrt{386}.

$$\frac{\sqrt{3}\sqrt{386}\sqrt{3}}{2\times 3\times 3}$$

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

$$\frac{3\sqrt{386}}{2\times 3\times 3}$$

Multiply 2 and 3 to get 6.

$$\frac{3\sqrt{386}}{6\times 3}$$

Multiply 6 and 3 to get 18.

$$\frac{3\sqrt{386}}{18}$$

Divide 3\sqrt{386} by 18 to get \frac{1}{6}\sqrt{386}.

$$\frac{1}{6}\sqrt{386}$$