Solve sqrt{48}+sqrt{1/8}-2sqrt{1/3}+sqrt{18}

$$\sqrt { 48 } + \sqrt { \frac { 1 } { 8 } } - 2 \sqrt { \frac { 1 } { 3 } } + \sqrt { 18 }$$

Evaluate

$$\frac{10\sqrt{3}}{3}+\frac{13\sqrt{2}}{4}\approx 10.36969677$$

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}.

$$4\sqrt{3}+\sqrt{\frac{1}{8}}-2\sqrt{\frac{1}{3}}+\sqrt{18}$$

Rewrite the square root of the division \sqrt{\frac{1}{8}} as the division of square roots \frac{\sqrt{1}}{\sqrt{8}}.

$$4\sqrt{3}+\frac{\sqrt{1}}{\sqrt{8}}-2\sqrt{\frac{1}{3}}+\sqrt{18}$$

Calculate the square root of 1 and get 1.

$$4\sqrt{3}+\frac{1}{\sqrt{8}}-2\sqrt{\frac{1}{3}}+\sqrt{18}$$

Factor 8=2^{2}\times 2. Rewrite the square root of the product \sqrt{2^{2}\times 2} as the product of square roots \sqrt{2^{2}}\sqrt{2}. Take the square root of 2^{2}.

$$4\sqrt{3}+\frac{1}{2\sqrt{2}}-2\sqrt{\frac{1}{3}}+\sqrt{18}$$

Rationalize the denominator of \frac{1}{2\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$4\sqrt{3}+\frac{\sqrt{2}}{2\left(\sqrt{2}\right)^{2}}-2\sqrt{\frac{1}{3}}+\sqrt{18}$$

The square of \sqrt{2} is 2.

$$4\sqrt{3}+\frac{\sqrt{2}}{2\times 2}-2\sqrt{\frac{1}{3}}+\sqrt{18}$$

Multiply 2 and 2 to get 4.

$$4\sqrt{3}+\frac{\sqrt{2}}{4}-2\sqrt{\frac{1}{3}}+\sqrt{18}$$

Rewrite the square root of the division \sqrt{\frac{1}{3}} as the division of square roots \frac{\sqrt{1}}{\sqrt{3}}.

$$4\sqrt{3}+\frac{\sqrt{2}}{4}-2\times \frac{\sqrt{1}}{\sqrt{3}}+\sqrt{18}$$

Calculate the square root of 1 and get 1.

$$4\sqrt{3}+\frac{\sqrt{2}}{4}-2\times \frac{1}{\sqrt{3}}+\sqrt{18}$$

Rationalize the denominator of \frac{1}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$4\sqrt{3}+\frac{\sqrt{2}}{4}-2\times \frac{\sqrt{3}}{\left(\sqrt{3}\right)^{2}}+\sqrt{18}$$

The square of \sqrt{3} is 3.

$$4\sqrt{3}+\frac{\sqrt{2}}{4}-2\times \frac{\sqrt{3}}{3}+\sqrt{18}$$

Express -2\times \frac{\sqrt{3}}{3} as a single fraction.

$$4\sqrt{3}+\frac{\sqrt{2}}{4}+\frac{-2\sqrt{3}}{3}+\sqrt{18}$$

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}.

$$4\sqrt{3}+\frac{\sqrt{2}}{4}+\frac{-2\sqrt{3}}{3}+3\sqrt{2}$$

Combine \frac{\sqrt{2}}{4} and 3\sqrt{2} to get \frac{13}{4}\sqrt{2}.

$$4\sqrt{3}+\frac{13}{4}\sqrt{2}+\frac{-2\sqrt{3}}{3}$$

Combine 4\sqrt{3} and \frac{-2\sqrt{3}}{3} to get \frac{10}{3}\sqrt{3}.

$$\frac{10}{3}\sqrt{3}+\frac{13}{4}\sqrt{2}$$