Solve sqrt{7^{4}*7^{2}+4^{2}+4^{3}-2^{3}}+sqrt{21-2*5-2}*sqrt{(2^2)^3+3^2-6*2^3}

$$\sqrt { 7 ^ { 4 } \cdot 7 ^ { 2 } + 4 ^ { 2 } + 4 ^ { 3 } - 2 ^ { 3 } } + \sqrt { 21 - 2 \cdot 5 - 2 } \cdot \sqrt { ( 2 ^ { 2 } ) ^ { 3 } + 3 ^ { 2 } - 6 \cdot 2 ^ { 3 } }$$

Evaluate

$$\sqrt{117721}+15\approx 358.104940215$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 4 and 2 to get 6.

$$\sqrt{7^{6}+4^{2}+4^{3}-2^{3}}+\sqrt{21-2\times 5-2}\sqrt{\left(2^{2}\right)^{3}+3^{2}-6\times 2^{3}}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 3 to get 6.

$$\sqrt{7^{6}+4^{2}+4^{3}-2^{3}}+\sqrt{21-2\times 5-2}\sqrt{2^{6}+3^{2}-6\times 2^{3}}$$

Calculate 7 to the power of 6 and get 117649.

$$\sqrt{117649+4^{2}+4^{3}-2^{3}}+\sqrt{21-2\times 5-2}\sqrt{2^{6}+3^{2}-6\times 2^{3}}$$

Calculate 4 to the power of 2 and get 16.

$$\sqrt{117649+16+4^{3}-2^{3}}+\sqrt{21-2\times 5-2}\sqrt{2^{6}+3^{2}-6\times 2^{3}}$$

Add 117649 and 16 to get 117665.

$$\sqrt{117665+4^{3}-2^{3}}+\sqrt{21-2\times 5-2}\sqrt{2^{6}+3^{2}-6\times 2^{3}}$$

Calculate 4 to the power of 3 and get 64.

$$\sqrt{117665+64-2^{3}}+\sqrt{21-2\times 5-2}\sqrt{2^{6}+3^{2}-6\times 2^{3}}$$

Add 117665 and 64 to get 117729.

$$\sqrt{117729-2^{3}}+\sqrt{21-2\times 5-2}\sqrt{2^{6}+3^{2}-6\times 2^{3}}$$

Calculate 2 to the power of 3 and get 8.

$$\sqrt{117729-8}+\sqrt{21-2\times 5-2}\sqrt{2^{6}+3^{2}-6\times 2^{3}}$$

Subtract 8 from 117729 to get 117721.

$$\sqrt{117721}+\sqrt{21-2\times 5-2}\sqrt{2^{6}+3^{2}-6\times 2^{3}}$$

Multiply 2 and 5 to get 10.

$$\sqrt{117721}+\sqrt{21-10-2}\sqrt{2^{6}+3^{2}-6\times 2^{3}}$$

Subtract 10 from 21 to get 11.

$$\sqrt{117721}+\sqrt{11-2}\sqrt{2^{6}+3^{2}-6\times 2^{3}}$$

Subtract 2 from 11 to get 9.

$$\sqrt{117721}+\sqrt{9}\sqrt{2^{6}+3^{2}-6\times 2^{3}}$$

Calculate the square root of 9 and get 3.

$$\sqrt{117721}+3\sqrt{2^{6}+3^{2}-6\times 2^{3}}$$

Calculate 2 to the power of 6 and get 64.

$$\sqrt{117721}+3\sqrt{64+3^{2}-6\times 2^{3}}$$

Calculate 3 to the power of 2 and get 9.

$$\sqrt{117721}+3\sqrt{64+9-6\times 2^{3}}$$

Add 64 and 9 to get 73.

$$\sqrt{117721}+3\sqrt{73-6\times 2^{3}}$$

Calculate 2 to the power of 3 and get 8.

$$\sqrt{117721}+3\sqrt{73-6\times 8}$$

Multiply 6 and 8 to get 48.

$$\sqrt{117721}+3\sqrt{73-48}$$

Subtract 48 from 73 to get 25.

$$\sqrt{117721}+3\sqrt{25}$$

Calculate the square root of 25 and get 5.

$$\sqrt{117721}+3\times 5$$

Multiply 3 and 5 to get 15.

$$\sqrt{117721}+15$$