Solve sqrt{32}*sqrt{1/2}+sqrt{20}

$$\sqrt { 32 } \times \sqrt { \frac { 1 } { 2 } } + \sqrt { 20 }$$

Evaluate

$$2\left(\sqrt{5}+2\right)\approx 8.472135955$$

Solution Steps

Factor 32=4^{2}\times 2. Rewrite the square root of the product \sqrt{4^{2}\times 2} as the product of square roots \sqrt{4^{2}}\sqrt{2}. Take the square root of 4^{2}.

$$4\sqrt{2}\sqrt{\frac{1}{2}}+\sqrt{20}$$

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

$$4\sqrt{2}\times \frac{\sqrt{1}}{\sqrt{2}}+\sqrt{20}$$

Calculate the square root of 1 and get 1.

$$4\sqrt{2}\times \frac{1}{\sqrt{2}}+\sqrt{20}$$

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

$$4\sqrt{2}\times \frac{\sqrt{2}}{\left(\sqrt{2}\right)^{2}}+\sqrt{20}$$

The square of \sqrt{2} is 2.

$$4\sqrt{2}\times \frac{\sqrt{2}}{2}+\sqrt{20}$$

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

$$2\sqrt{2}\sqrt{2}+\sqrt{20}$$

Factor 20=2^{2}\times 5. Rewrite the square root of the product \sqrt{2^{2}\times 5} as the product of square roots \sqrt{2^{2}}\sqrt{5}. Take the square root of 2^{2}.

$$2\sqrt{2}\sqrt{2}+2\sqrt{5}$$

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

$$2\times 2+2\sqrt{5}$$

Multiply 2 and 2 to get 4.

$$4+2\sqrt{5}$$