Solve sqrt{18}-4sqrt{1/2}+sqrt{24}+sqrt{3}

$$\sqrt { 18 } - 4 \sqrt { \frac { 1 } { 2 } } + \sqrt { 24 } + \sqrt { 3 }$$

Evaluate

$$\sqrt{2}+\sqrt{3}+2\sqrt{6}\approx 8.045243856$$

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}.

$$3\sqrt{2}-4\sqrt{\frac{1}{2}}+\sqrt{24}+\sqrt{3}$$

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

$$3\sqrt{2}-4\times \frac{\sqrt{1}}{\sqrt{2}}+\sqrt{24}+\sqrt{3}$$

Calculate the square root of 1 and get 1.

$$3\sqrt{2}-4\times \frac{1}{\sqrt{2}}+\sqrt{24}+\sqrt{3}$$

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

$$3\sqrt{2}-4\times \frac{\sqrt{2}}{\left(\sqrt{2}\right)^{2}}+\sqrt{24}+\sqrt{3}$$

The square of \sqrt{2} is 2.

$$3\sqrt{2}-4\times \frac{\sqrt{2}}{2}+\sqrt{24}+\sqrt{3}$$

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

$$3\sqrt{2}-2\sqrt{2}+\sqrt{24}+\sqrt{3}$$

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

$$\sqrt{2}+\sqrt{24}+\sqrt{3}$$

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}.

$$\sqrt{2}+2\sqrt{6}+\sqrt{3}$$