Solve sqrt{24}*sqrt{1/2}+sqrt{3}

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

Evaluate

$$3\sqrt{3}\approx 5.196152423$$

Solution Steps

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}.

$$2\sqrt{6}\sqrt{\frac{1}{2}}+\sqrt{3}$$

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

$$2\sqrt{6}\times \frac{\sqrt{1}}{\sqrt{2}}+\sqrt{3}$$

Calculate the square root of 1 and get 1.

$$2\sqrt{6}\times \frac{1}{\sqrt{2}}+\sqrt{3}$$

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

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

The square of \sqrt{2} is 2.

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

Cancel out 2 and 2.

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

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}.

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

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

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

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

$$3\sqrt{3}$$