Solve sqrt{32}*frac{sqrt{8}*4}{sqrt{2}}=

$$\sqrt { 32 } \cdot \frac { \sqrt { 8 } \cdot 4 } { \sqrt { 2 } } =$$

Evaluate

$$32\sqrt{2}\approx 45.254833996$$

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}.

$$4\sqrt{2}\times \frac{4\sqrt{8}}{\sqrt{2}}$$

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}.

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

Multiply 4 and 2 to get 8.

$$4\sqrt{2}\times \frac{8\sqrt{2}}{\sqrt{2}}$$

Rationalize the denominator of \frac{8\sqrt{2}}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$4\sqrt{2}\times \frac{8\sqrt{2}\sqrt{2}}{\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$4\sqrt{2}\times \frac{8\sqrt{2}\sqrt{2}}{2}$$

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

$$4\sqrt{2}\times \frac{8\times 2}{2}$$

Multiply 8 and 2 to get 16.

$$4\sqrt{2}\times \frac{16}{2}$$

Divide 16 by 2 to get 8.

$$4\sqrt{2}\times 8$$

Multiply 4 and 8 to get 32.

$$32\sqrt{2}$$