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

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

Evaluate

$$3\sqrt{2}\approx 4.242640687$$

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}.

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

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

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

Calculate the square root of 1 and get 1.

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

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

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

The square of \sqrt{2} is 2.

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

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

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

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

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

Rewrite the division of square roots \frac{\sqrt{24}}{\sqrt{3}} as the square root of the division \sqrt{\frac{24}{3}} and perform the division.

$$\sqrt{2}+\sqrt{8}$$

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}.

$$\sqrt{2}+2\sqrt{2}$$

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

$$3\sqrt{2}$$