Solve (sqrt{48}+1/2sqrt{3/2})divsqrt{27}

$$( \sqrt{ 48 } + \frac{ 1 }{ 2 } \sqrt{ \frac{ 3 }{ 2 } } ) \div \sqrt{ 27 }$$

Evaluate

$$\frac{\sqrt{2}}{12}+\frac{4}{3}\approx 1.451184464$$

Short Solution Steps

Factor 48=4^{2}\times 3. Rewrite the square root of the product \sqrt{4^{2}\times 3} as the product of square roots \sqrt{4^{2}}\sqrt{3}. Take the square root of 4^{2}.

$$\frac{4\sqrt{3}+\frac{1}{2}\sqrt{\frac{3}{2}}}{\sqrt{27}}$$

Rewrite the square root of the division \sqrt{\frac{3}{2}} as the division of square roots \frac{\sqrt{3}}{\sqrt{2}}.

$$\frac{4\sqrt{3}+\frac{1}{2}\times \frac{\sqrt{3}}{\sqrt{2}}}{\sqrt{27}}$$

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

$$\frac{4\sqrt{3}+\frac{1}{2}\times \frac{\sqrt{3}\sqrt{2}}{\left(\sqrt{2}\right)^{2}}}{\sqrt{27}}$$

The square of \sqrt{2} is 2.

$$\frac{4\sqrt{3}+\frac{1}{2}\times \frac{\sqrt{3}\sqrt{2}}{2}}{\sqrt{27}}$$

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

$$\frac{4\sqrt{3}+\frac{1}{2}\times \frac{\sqrt{6}}{2}}{\sqrt{27}}$$

Multiply \frac{1}{2} times \frac{\sqrt{6}}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{4\sqrt{3}+\frac{\sqrt{6}}{2\times 2}}{\sqrt{27}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 4\sqrt{3} times \frac{2\times 2}{2\times 2}.

$$\frac{\frac{4\sqrt{3}\times 2\times 2}{2\times 2}+\frac{\sqrt{6}}{2\times 2}}{\sqrt{27}}$$

Since \frac{4\sqrt{3}\times 2\times 2}{2\times 2} and \frac{\sqrt{6}}{2\times 2} have the same denominator, add them by adding their numerators.

$$\frac{\frac{4\sqrt{3}\times 2\times 2+\sqrt{6}}{2\times 2}}{\sqrt{27}}$$

Do the multiplications in 4\sqrt{3}\times 2\times 2+\sqrt{6}.

$$\frac{\frac{16\sqrt{3}+\sqrt{6}}{2\times 2}}{\sqrt{27}}$$

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}.

$$\frac{\frac{16\sqrt{3}+\sqrt{6}}{2\times 2}}{3\sqrt{3}}$$

Express \frac{\frac{16\sqrt{3}+\sqrt{6}}{2\times 2}}{3\sqrt{3}} as a single fraction.

$$\frac{16\sqrt{3}+\sqrt{6}}{2\times 2\times 3\sqrt{3}}$$

Rationalize the denominator of \frac{16\sqrt{3}+\sqrt{6}}{2\times 2\times 3\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{\left(16\sqrt{3}+\sqrt{6}\right)\sqrt{3}}{2\times 2\times 3\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{\left(16\sqrt{3}+\sqrt{6}\right)\sqrt{3}}{2\times 2\times 3\times 3}$$

Multiply 2 and 2 to get 4.

$$\frac{\left(16\sqrt{3}+\sqrt{6}\right)\sqrt{3}}{4\times 3\times 3}$$

Multiply 4 and 3 to get 12.

$$\frac{\left(16\sqrt{3}+\sqrt{6}\right)\sqrt{3}}{12\times 3}$$

Multiply 12 and 3 to get 36.

$$\frac{\left(16\sqrt{3}+\sqrt{6}\right)\sqrt{3}}{36}$$

Use the distributive property to multiply 16\sqrt{3}+\sqrt{6} by \sqrt{3}.

$$\frac{16\left(\sqrt{3}\right)^{2}+\sqrt{6}\sqrt{3}}{36}$$

The square of \sqrt{3} is 3.

$$\frac{16\times 3+\sqrt{6}\sqrt{3}}{36}$$

Multiply 16 and 3 to get 48.

$$\frac{48+\sqrt{6}\sqrt{3}}{36}$$

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}.

$$\frac{48+\sqrt{3}\sqrt{2}\sqrt{3}}{36}$$

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

$$\frac{48+3\sqrt{2}}{36}$$

Factor

$$\frac{\sqrt{2} + 16}{12} = 1.4511844635310913$$