Solve sqrt{sqrt{4}*5*sqrt{25}}*sqrt{sqrt{2}*sqrt{8}}

$$\sqrt { \sqrt { 4 } \cdot 5 \cdot \sqrt { 25 } } \cdot \sqrt { \sqrt { 2 } \cdot \sqrt { 8 } }$$

Evaluate

$$10\sqrt{2}\approx 14.142135624$$

Solution Steps

Calculate the square root of 4 and get 2.

$$\sqrt{5\times 2\sqrt{25}}\sqrt{\sqrt{2}\sqrt{8}}$$

Multiply 5 and 2 to get 10.

$$\sqrt{10\sqrt{25}}\sqrt{\sqrt{2}\sqrt{8}}$$

Calculate the square root of 25 and get 5.

$$\sqrt{10\times 5}\sqrt{\sqrt{2}\sqrt{8}}$$

Multiply 10 and 5 to get 50.

$$\sqrt{50}\sqrt{\sqrt{2}\sqrt{8}}$$

Factor 50=5^{2}\times 2. Rewrite the square root of the product \sqrt{5^{2}\times 2} as the product of square roots \sqrt{5^{2}}\sqrt{2}. Take the square root of 5^{2}.

$$5\sqrt{2}\sqrt{\sqrt{2}\sqrt{8}}$$

Factor 8=2\times 4. Rewrite the square root of the product \sqrt{2\times 4} as the product of square roots \sqrt{2}\sqrt{4}.

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

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

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

Calculate the square root of 4 and get 2.

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

Multiply 2 and 2 to get 4.

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

Calculate the square root of 4 and get 2.

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

Multiply 5 and 2 to get 10.

$$10\sqrt{2}$$