Solve sqrt{48}divsqrt{3}-sqrt{1/2}*sqrt{54}+sqrt{24}divfrac{sqrt{2}}{2}

$$\sqrt { 48 } \div \sqrt { 3 } - \sqrt { \frac { 1 } { 2 } } \times \sqrt { 54 } + \sqrt { 24 } \div \frac { \sqrt { 2 } } { 2 }$$

Evaluate

$$\sqrt{3}+4\approx 5.732050808$$

Solution Steps

Rewrite the division of square roots \frac{\sqrt{48}}{\sqrt{3}} as the square root of the division \sqrt{\frac{48}{3}} and perform the division.

$$\sqrt{16}-\sqrt{\frac{1}{2}}\sqrt{54}+\frac{\sqrt{24}}{\frac{\sqrt{2}}{2}}$$

Calculate the square root of 16 and get 4.

$$4-\sqrt{\frac{1}{2}}\sqrt{54}+\frac{\sqrt{24}}{\frac{\sqrt{2}}{2}}$$

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

$$4-\frac{\sqrt{1}}{\sqrt{2}}\sqrt{54}+\frac{\sqrt{24}}{\frac{\sqrt{2}}{2}}$$

Calculate the square root of 1 and get 1.

$$4-\frac{1}{\sqrt{2}}\sqrt{54}+\frac{\sqrt{24}}{\frac{\sqrt{2}}{2}}$$

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

$$4-\frac{\sqrt{2}}{\left(\sqrt{2}\right)^{2}}\sqrt{54}+\frac{\sqrt{24}}{\frac{\sqrt{2}}{2}}$$

The square of \sqrt{2} is 2.

$$4-\frac{\sqrt{2}}{2}\sqrt{54}+\frac{\sqrt{24}}{\frac{\sqrt{2}}{2}}$$

Factor 54=3^{2}\times 6. Rewrite the square root of the product \sqrt{3^{2}\times 6} as the product of square roots \sqrt{3^{2}}\sqrt{6}. Take the square root of 3^{2}.

$$4-\frac{\sqrt{2}}{2}\times 3\sqrt{6}+\frac{\sqrt{24}}{\frac{\sqrt{2}}{2}}$$

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

$$4-\frac{\sqrt{2}\times 3}{2}\sqrt{6}+\frac{\sqrt{24}}{\frac{\sqrt{2}}{2}}$$

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

$$4-\frac{\sqrt{2}\times 3\sqrt{6}}{2}+\frac{\sqrt{24}}{\frac{\sqrt{2}}{2}}$$

Factor 6=2\times 3. Rewrite the square root of the product \sqrt{2\times 3} as the product of square roots \sqrt{2}\sqrt{3}.

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

Multiply \sqrt{2} and \sqrt{2} to get 2.

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

Multiply 2 and 3 to get 6.

$$4-\frac{6\sqrt{3}}{2}+\frac{\sqrt{24}}{\frac{\sqrt{2}}{2}}$$

Divide 6\sqrt{3} by 2 to get 3\sqrt{3}.

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

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}.

$$4-3\sqrt{3}+\frac{2\sqrt{6}}{\frac{\sqrt{2}}{2}}$$

Divide 2\sqrt{6} by \frac{\sqrt{2}}{2} by multiplying 2\sqrt{6} by the reciprocal of \frac{\sqrt{2}}{2}.

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

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

$$4-3\sqrt{3}+\frac{2\sqrt{6}\times 2\sqrt{2}}{\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

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

Multiply 2 and 2 to get 4.

$$4-3\sqrt{3}+\frac{4\sqrt{6}\sqrt{2}}{2}$$

Factor 6=2\times 3. Rewrite the square root of the product \sqrt{2\times 3} as the product of square roots \sqrt{2}\sqrt{3}.

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

Multiply \sqrt{2} and \sqrt{2} to get 2.

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

Cancel out 2 and 2.

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

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

$$4+\sqrt{3}$$