Solve sqrt[3]{1+3(2^{2}+1)(2^{4}+1)(2^8+1)(2^16+1)(2^32+1)(2^64+1)}

$$\sqrt[ 3 ] { 1 + 3 ( 2 ^ { 2 } + 1 ) ( 2 ^ { 4 } + 1 ) ( 2 ^ { 8 } + 1 ) ( 2 ^ { 16 } + 1 ) ( 2 ^ { 32 } + 1 ) ( 2 ^ { 64 } + 1 ) }$$

Evaluate

$$4398046511104\sqrt[3]{4}\approx 6.981463658 \cdot 10^{12}$$

Solution Steps

Calculate 2 to the power of 2 and get 4.

$$\sqrt[3]{1+3\left(4+1\right)\left(2^{4}+1\right)\left(2^{8}+1\right)\left(2^{16}+1\right)\left(2^{32}+1\right)\left(2^{64}+1\right)}$$

Add 4 and 1 to get 5.

$$\sqrt[3]{1+3\times 5\left(2^{4}+1\right)\left(2^{8}+1\right)\left(2^{16}+1\right)\left(2^{32}+1\right)\left(2^{64}+1\right)}$$

Multiply 3 and 5 to get 15.

$$\sqrt[3]{1+15\left(2^{4}+1\right)\left(2^{8}+1\right)\left(2^{16}+1\right)\left(2^{32}+1\right)\left(2^{64}+1\right)}$$

Calculate 2 to the power of 4 and get 16.

$$\sqrt[3]{1+15\left(16+1\right)\left(2^{8}+1\right)\left(2^{16}+1\right)\left(2^{32}+1\right)\left(2^{64}+1\right)}$$

Add 16 and 1 to get 17.

$$\sqrt[3]{1+15\times 17\left(2^{8}+1\right)\left(2^{16}+1\right)\left(2^{32}+1\right)\left(2^{64}+1\right)}$$

Multiply 15 and 17 to get 255.

$$\sqrt[3]{1+255\left(2^{8}+1\right)\left(2^{16}+1\right)\left(2^{32}+1\right)\left(2^{64}+1\right)}$$

Calculate 2 to the power of 8 and get 256.

$$\sqrt[3]{1+255\left(256+1\right)\left(2^{16}+1\right)\left(2^{32}+1\right)\left(2^{64}+1\right)}$$

Add 256 and 1 to get 257.

$$\sqrt[3]{1+255\times 257\left(2^{16}+1\right)\left(2^{32}+1\right)\left(2^{64}+1\right)}$$

Multiply 255 and 257 to get 65535.

$$\sqrt[3]{1+65535\left(2^{16}+1\right)\left(2^{32}+1\right)\left(2^{64}+1\right)}$$

Calculate 2 to the power of 16 and get 65536.

$$\sqrt[3]{1+65535\left(65536+1\right)\left(2^{32}+1\right)\left(2^{64}+1\right)}$$

Add 65536 and 1 to get 65537.

$$\sqrt[3]{1+65535\times 65537\left(2^{32}+1\right)\left(2^{64}+1\right)}$$

Multiply 65535 and 65537 to get 4294967295.

$$\sqrt[3]{1+4294967295\left(2^{32}+1\right)\left(2^{64}+1\right)}$$

Calculate 2 to the power of 32 and get 4294967296.

$$\sqrt[3]{1+4294967295\left(4294967296+1\right)\left(2^{64}+1\right)}$$

Add 4294967296 and 1 to get 4294967297.

$$\sqrt[3]{1+4294967295\times 4294967297\left(2^{64}+1\right)}$$

Multiply 4294967295 and 4294967297 to get 18446744073709551615.

$$\sqrt[3]{1+18446744073709551615\left(2^{64}+1\right)}$$

Calculate 2 to the power of 64 and get 18446744073709551616.

$$\sqrt[3]{1+18446744073709551615\left(18446744073709551616+1\right)}$$

Add 18446744073709551616 and 1 to get 18446744073709551617.

$$\sqrt[3]{1+18446744073709551615\times 18446744073709551617}$$

Multiply 18446744073709551615 and 18446744073709551617 to get 340282366920938463463374607431768211455.

$$\sqrt[3]{1+340282366920938463463374607431768211455}$$

Add 1 and 340282366920938463463374607431768211455 to get 340282366920938463463374607431768211456.

$$\sqrt[3]{340282366920938463463374607431768211456}$$