Solve sqrt{18}+4sqrt{1/2}-sqrt{48}+sqrt{1/27}

$$\sqrt{ 18 } +4 \sqrt{ \frac{ 1 }{ 2 } } - \sqrt{ 48 } + \sqrt{ \frac{ 1 }{ 27 } }$$

Evaluate

$$-\frac{35\sqrt{3}}{9}+5\sqrt{2}\approx 0.335314671$$

Solution Steps

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}.

$$3\sqrt{2}+4\sqrt{\frac{1}{2}}-\sqrt{48}+\sqrt{\frac{1}{27}}$$

Rewrite the square root of the division \sqrt{\frac{1}{2}} as the division of square roots \frac{\sqrt{1}}{\sqrt{2}}.

$$3\sqrt{2}+4\times \frac{\sqrt{1}}{\sqrt{2}}-\sqrt{48}+\sqrt{\frac{1}{27}}$$

Calculate the square root of 1 and get 1.

$$3\sqrt{2}+4\times \frac{1}{\sqrt{2}}-\sqrt{48}+\sqrt{\frac{1}{27}}$$

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

$$3\sqrt{2}+4\times \frac{\sqrt{2}}{\left(\sqrt{2}\right)^{2}}-\sqrt{48}+\sqrt{\frac{1}{27}}$$

The square of \sqrt{2} is 2.

$$3\sqrt{2}+4\times \frac{\sqrt{2}}{2}-\sqrt{48}+\sqrt{\frac{1}{27}}$$

Cancel out 2, the greatest common factor in 4 and 2.

$$3\sqrt{2}+2\sqrt{2}-\sqrt{48}+\sqrt{\frac{1}{27}}$$

Combine 3\sqrt{2} and 2\sqrt{2} to get 5\sqrt{2}.

$$5\sqrt{2}-\sqrt{48}+\sqrt{\frac{1}{27}}$$

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}.

$$5\sqrt{2}-4\sqrt{3}+\sqrt{\frac{1}{27}}$$

Rewrite the square root of the division \sqrt{\frac{1}{27}} as the division of square roots \frac{\sqrt{1}}{\sqrt{27}}.

$$5\sqrt{2}-4\sqrt{3}+\frac{\sqrt{1}}{\sqrt{27}}$$

Calculate the square root of 1 and get 1.

$$5\sqrt{2}-4\sqrt{3}+\frac{1}{\sqrt{27}}$$

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}.

$$5\sqrt{2}-4\sqrt{3}+\frac{1}{3\sqrt{3}}$$

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

$$5\sqrt{2}-4\sqrt{3}+\frac{\sqrt{3}}{3\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$5\sqrt{2}-4\sqrt{3}+\frac{\sqrt{3}}{3\times 3}$$

Multiply 3 and 3 to get 9.

$$5\sqrt{2}-4\sqrt{3}+\frac{\sqrt{3}}{9}$$

Combine -4\sqrt{3} and \frac{\sqrt{3}}{9} to get -\frac{35}{9}\sqrt{3}.

$$5\sqrt{2}-\frac{35}{9}\sqrt{3}$$