Solve sqrt{2^{9}}:sqrt{2}:{(6*sqrt{2^8}+sqrt{9}*8):5-[12*sqrt{25}-(12^2-2^2*15):3-sqrt{4^4}]:2}

$$\sqrt { 2 ^ { 9 } } : \sqrt { 2 } : \{ ( 6 \times \sqrt { 2 ^ { 8 } } + \sqrt { 9 } \times 8 ) : 5 - [ 12 \times \sqrt { 25 } - ( 12 ^ { 2 } - 2 ^ { 2 } \times 15 ) : 3 - \sqrt { 4 ^ { 4 } } ] : 2 \}$$

Evaluate

$$1$$

Solution Steps

Calculate 2 to the power of 9 and get 512.

$$\frac{\frac{\sqrt{512}}{\sqrt{2}}}{\frac{6\sqrt{2^{8}}+8\sqrt{9}}{5}-\frac{12\sqrt{25}-\frac{12^{2}-2^{2}\times 15}{3}-\sqrt{4^{4}}}{2}}$$

Factor 512=16^{2}\times 2. Rewrite the square root of the product \sqrt{16^{2}\times 2} as the product of square roots \sqrt{16^{2}}\sqrt{2}. Take the square root of 16^{2}.

$$\frac{\frac{16\sqrt{2}}{\sqrt{2}}}{\frac{6\sqrt{2^{8}}+8\sqrt{9}}{5}-\frac{12\sqrt{25}-\frac{12^{2}-2^{2}\times 15}{3}-\sqrt{4^{4}}}{2}}$$

Rationalize the denominator of \frac{16\sqrt{2}}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{\frac{16\sqrt{2}\sqrt{2}}{\left(\sqrt{2}\right)^{2}}}{\frac{6\sqrt{2^{8}}+8\sqrt{9}}{5}-\frac{12\sqrt{25}-\frac{12^{2}-2^{2}\times 15}{3}-\sqrt{4^{4}}}{2}}$$

The square of \sqrt{2} is 2.

$$\frac{\frac{16\sqrt{2}\sqrt{2}}{2}}{\frac{6\sqrt{2^{8}}+8\sqrt{9}}{5}-\frac{12\sqrt{25}-\frac{12^{2}-2^{2}\times 15}{3}-\sqrt{4^{4}}}{2}}$$

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

$$\frac{\frac{16\times 2}{2}}{\frac{6\sqrt{2^{8}}+8\sqrt{9}}{5}-\frac{12\sqrt{25}-\frac{12^{2}-2^{2}\times 15}{3}-\sqrt{4^{4}}}{2}}$$

Multiply 16 and 2 to get 32.

$$\frac{\frac{32}{2}}{\frac{6\sqrt{2^{8}}+8\sqrt{9}}{5}-\frac{12\sqrt{25}-\frac{12^{2}-2^{2}\times 15}{3}-\sqrt{4^{4}}}{2}}$$

Divide 32 by 2 to get 16.

$$\frac{16}{\frac{6\sqrt{2^{8}}+8\sqrt{9}}{5}-\frac{12\sqrt{25}-\frac{12^{2}-2^{2}\times 15}{3}-\sqrt{4^{4}}}{2}}$$

Calculate 2 to the power of 8 and get 256.

$$\frac{16}{\frac{6\sqrt{256}+8\sqrt{9}}{5}-\frac{12\sqrt{25}-\frac{12^{2}-2^{2}\times 15}{3}-\sqrt{4^{4}}}{2}}$$

Calculate the square root of 256 and get 16.

$$\frac{16}{\frac{6\times 16+8\sqrt{9}}{5}-\frac{12\sqrt{25}-\frac{12^{2}-2^{2}\times 15}{3}-\sqrt{4^{4}}}{2}}$$

Multiply 6 and 16 to get 96.

$$\frac{16}{\frac{96+8\sqrt{9}}{5}-\frac{12\sqrt{25}-\frac{12^{2}-2^{2}\times 15}{3}-\sqrt{4^{4}}}{2}}$$

Calculate the square root of 9 and get 3.

$$\frac{16}{\frac{96+8\times 3}{5}-\frac{12\sqrt{25}-\frac{12^{2}-2^{2}\times 15}{3}-\sqrt{4^{4}}}{2}}$$

Multiply 8 and 3 to get 24.

$$\frac{16}{\frac{96+24}{5}-\frac{12\sqrt{25}-\frac{12^{2}-2^{2}\times 15}{3}-\sqrt{4^{4}}}{2}}$$

Add 96 and 24 to get 120.

$$\frac{16}{\frac{120}{5}-\frac{12\sqrt{25}-\frac{12^{2}-2^{2}\times 15}{3}-\sqrt{4^{4}}}{2}}$$

Divide 120 by 5 to get 24.

$$\frac{16}{24-\frac{12\sqrt{25}-\frac{12^{2}-2^{2}\times 15}{3}-\sqrt{4^{4}}}{2}}$$

Calculate the square root of 25 and get 5.

$$\frac{16}{24-\frac{12\times 5-\frac{12^{2}-2^{2}\times 15}{3}-\sqrt{4^{4}}}{2}}$$

Multiply 12 and 5 to get 60.

$$\frac{16}{24-\frac{60-\frac{12^{2}-2^{2}\times 15}{3}-\sqrt{4^{4}}}{2}}$$

Calculate 12 to the power of 2 and get 144.

$$\frac{16}{24-\frac{60-\frac{144-2^{2}\times 15}{3}-\sqrt{4^{4}}}{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{16}{24-\frac{60-\frac{144-4\times 15}{3}-\sqrt{4^{4}}}{2}}$$

Multiply 4 and 15 to get 60.

$$\frac{16}{24-\frac{60-\frac{144-60}{3}-\sqrt{4^{4}}}{2}}$$

Subtract 60 from 144 to get 84.

$$\frac{16}{24-\frac{60-\frac{84}{3}-\sqrt{4^{4}}}{2}}$$

Divide 84 by 3 to get 28.

$$\frac{16}{24-\frac{60-28-\sqrt{4^{4}}}{2}}$$

Subtract 28 from 60 to get 32.

$$\frac{16}{24-\frac{32-\sqrt{4^{4}}}{2}}$$

Calculate 4 to the power of 4 and get 256.

$$\frac{16}{24-\frac{32-\sqrt{256}}{2}}$$

Calculate the square root of 256 and get 16.

$$\frac{16}{24-\frac{32-16}{2}}$$

Subtract 16 from 32 to get 16.

$$\frac{16}{24-\frac{16}{2}}$$

Divide 16 by 2 to get 8.

$$\frac{16}{24-8}$$

Subtract 8 from 24 to get 16.

$$\frac{16}{16}$$

Divide 16 by 16 to get 1.

$$1$$

Factor

$$1$$