Solve {left(sqrt{12}+sqrt{6}+sqrt{26}right)}^2

$${ \left( \sqrt{ 12 } + \sqrt{ 6 } + \sqrt{ 26 } \right) }^{ 2 }$$

Evaluate

$$4\left(\sqrt{39}+\sqrt{78}+3\sqrt{2}+11\right)\approx 121.277598207$$

Solution Steps

Factor 12=2^{2}\times 3. Rewrite the square root of the product \sqrt{2^{2}\times 3} as the product of square roots \sqrt{2^{2}}\sqrt{3}. Take the square root of 2^{2}.

$$\left(2\sqrt{3}+\sqrt{6}+\sqrt{26}\right)^{2}$$

Square 2\sqrt{3}+\sqrt{6}+\sqrt{26}.

$$2\sqrt{6}\sqrt{26}+4\sqrt{3}\sqrt{6}+4\sqrt{3}\sqrt{26}+4\left(\sqrt{3}\right)^{2}+\left(\sqrt{6}\right)^{2}+\left(\sqrt{26}\right)^{2}$$

To multiply \sqrt{6} and \sqrt{26}, multiply the numbers under the square root.

$$2\sqrt{156}+4\sqrt{3}\sqrt{6}+4\sqrt{3}\sqrt{26}+4\left(\sqrt{3}\right)^{2}+\left(\sqrt{6}\right)^{2}+\left(\sqrt{26}\right)^{2}$$

Factor 6=3\times 2. Rewrite the square root of the product \sqrt{3\times 2} as the product of square roots \sqrt{3}\sqrt{2}.

$$2\sqrt{156}+4\sqrt{3}\sqrt{3}\sqrt{2}+4\sqrt{3}\sqrt{26}+4\left(\sqrt{3}\right)^{2}+\left(\sqrt{6}\right)^{2}+\left(\sqrt{26}\right)^{2}$$

Multiply \sqrt{3} and \sqrt{3} to get 3.

$$2\sqrt{156}+4\times 3\sqrt{2}+4\sqrt{3}\sqrt{26}+4\left(\sqrt{3}\right)^{2}+\left(\sqrt{6}\right)^{2}+\left(\sqrt{26}\right)^{2}$$

Multiply 4 and 3 to get 12.

$$2\sqrt{156}+12\sqrt{2}+4\sqrt{3}\sqrt{26}+4\left(\sqrt{3}\right)^{2}+\left(\sqrt{6}\right)^{2}+\left(\sqrt{26}\right)^{2}$$

To multiply \sqrt{3} and \sqrt{26}, multiply the numbers under the square root.

$$2\sqrt{156}+12\sqrt{2}+4\sqrt{78}+4\left(\sqrt{3}\right)^{2}+\left(\sqrt{6}\right)^{2}+\left(\sqrt{26}\right)^{2}$$

The square of \sqrt{3} is 3.

$$2\sqrt{156}+12\sqrt{2}+4\sqrt{78}+4\times 3+\left(\sqrt{6}\right)^{2}+\left(\sqrt{26}\right)^{2}$$

Multiply 4 and 3 to get 12.

$$2\sqrt{156}+12\sqrt{2}+4\sqrt{78}+12+\left(\sqrt{6}\right)^{2}+\left(\sqrt{26}\right)^{2}$$

The square of \sqrt{6} is 6.

$$2\sqrt{156}+12\sqrt{2}+4\sqrt{78}+12+6+\left(\sqrt{26}\right)^{2}$$

Add 12 and 6 to get 18.

$$2\sqrt{156}+12\sqrt{2}+4\sqrt{78}+18+\left(\sqrt{26}\right)^{2}$$

The square of \sqrt{26} is 26.

$$2\sqrt{156}+12\sqrt{2}+4\sqrt{78}+18+26$$

Add 18 and 26 to get 44.

$$2\sqrt{156}+12\sqrt{2}+4\sqrt{78}+44$$

Factor 156=2^{2}\times 39. Rewrite the square root of the product \sqrt{2^{2}\times 39} as the product of square roots \sqrt{2^{2}}\sqrt{39}. Take the square root of 2^{2}.

$$2\times 2\sqrt{39}+12\sqrt{2}+4\sqrt{78}+44$$

Multiply 2 and 2 to get 4.

$$4\sqrt{39}+12\sqrt{2}+4\sqrt{78}+44$$

Expand

$$4 \sqrt{39} + 4 \sqrt{78} + 12 \sqrt{2} + 44 = 121.277598207$$

Solution Steps

Factor 12=2^{2}\times 3. Rewrite the square root of the product \sqrt{2^{2}\times 3} as the product of square roots \sqrt{2^{2}}\sqrt{3}. Take the square root of 2^{2}.

$$\left(2\sqrt{3}+\sqrt{6}+\sqrt{26}\right)^{2}$$

Square 2\sqrt{3}+\sqrt{6}+\sqrt{26}.

$$2\sqrt{6}\sqrt{26}+4\sqrt{3}\sqrt{6}+4\sqrt{3}\sqrt{26}+4\left(\sqrt{3}\right)^{2}+\left(\sqrt{6}\right)^{2}+\left(\sqrt{26}\right)^{2}$$

To multiply \sqrt{6} and \sqrt{26}, multiply the numbers under the square root.

$$2\sqrt{156}+4\sqrt{3}\sqrt{6}+4\sqrt{3}\sqrt{26}+4\left(\sqrt{3}\right)^{2}+\left(\sqrt{6}\right)^{2}+\left(\sqrt{26}\right)^{2}$$

Factor 6=3\times 2. Rewrite the square root of the product \sqrt{3\times 2} as the product of square roots \sqrt{3}\sqrt{2}.

$$2\sqrt{156}+4\sqrt{3}\sqrt{3}\sqrt{2}+4\sqrt{3}\sqrt{26}+4\left(\sqrt{3}\right)^{2}+\left(\sqrt{6}\right)^{2}+\left(\sqrt{26}\right)^{2}$$

Multiply \sqrt{3} and \sqrt{3} to get 3.

$$2\sqrt{156}+4\times 3\sqrt{2}+4\sqrt{3}\sqrt{26}+4\left(\sqrt{3}\right)^{2}+\left(\sqrt{6}\right)^{2}+\left(\sqrt{26}\right)^{2}$$

Multiply 4 and 3 to get 12.

$$2\sqrt{156}+12\sqrt{2}+4\sqrt{3}\sqrt{26}+4\left(\sqrt{3}\right)^{2}+\left(\sqrt{6}\right)^{2}+\left(\sqrt{26}\right)^{2}$$

To multiply \sqrt{3} and \sqrt{26}, multiply the numbers under the square root.

$$2\sqrt{156}+12\sqrt{2}+4\sqrt{78}+4\left(\sqrt{3}\right)^{2}+\left(\sqrt{6}\right)^{2}+\left(\sqrt{26}\right)^{2}$$

The square of \sqrt{3} is 3.

$$2\sqrt{156}+12\sqrt{2}+4\sqrt{78}+4\times 3+\left(\sqrt{6}\right)^{2}+\left(\sqrt{26}\right)^{2}$$

Multiply 4 and 3 to get 12.

$$2\sqrt{156}+12\sqrt{2}+4\sqrt{78}+12+\left(\sqrt{6}\right)^{2}+\left(\sqrt{26}\right)^{2}$$

The square of \sqrt{6} is 6.

$$2\sqrt{156}+12\sqrt{2}+4\sqrt{78}+12+6+\left(\sqrt{26}\right)^{2}$$

Add 12 and 6 to get 18.

$$2\sqrt{156}+12\sqrt{2}+4\sqrt{78}+18+\left(\sqrt{26}\right)^{2}$$

The square of \sqrt{26} is 26.

$$2\sqrt{156}+12\sqrt{2}+4\sqrt{78}+18+26$$

Add 18 and 26 to get 44.

$$2\sqrt{156}+12\sqrt{2}+4\sqrt{78}+44$$

Factor 156=2^{2}\times 39. Rewrite the square root of the product \sqrt{2^{2}\times 39} as the product of square roots \sqrt{2^{2}}\sqrt{39}. Take the square root of 2^{2}.

$$2\times 2\sqrt{39}+12\sqrt{2}+4\sqrt{78}+44$$

Multiply 2 and 2 to get 4.

$$4\sqrt{39}+12\sqrt{2}+4\sqrt{78}+44$$