Solve 50^2+[25(sqrt{6}-sqrt{2})]^2-2*50*2(sqrt{6}-sqrt{2})*(633)^2

$$50 ^ { 2 } + [ 25 ( \sqrt { 6 } - \sqrt { 2 } ) ] ^ { 2 } - 2 \times 50 \times 2 ( \sqrt { 6 } - \sqrt { 2 } ) \times ( 633 ) ^ { 2 }$$

Evaluate

$$80137800\sqrt{2}[phone]\sqrt{3}-80137800\sqrt{6}\approx -82961585.617486045$$

Solution Steps

Calculate 50 to the power of 2 and get 2500.

$$2500+\left(25\left(\sqrt{6}-\sqrt{2}\right)\right)^{2}-2\times 2\times 50\left(\sqrt{6}-\sqrt{2}\right)\times 633^{2}$$

Use the distributive property to multiply 25 by \sqrt{6}-\sqrt{2}.

$$2500+\left(25\sqrt{6}-25\sqrt{2}\right)^{2}-2\times 2\times 50\left(\sqrt{6}-\sqrt{2}\right)\times 633^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(25\sqrt{6}-25\sqrt{2}\right)^{2}.

$$2500+625\left(\sqrt{6}\right)^{2}-1250\sqrt{6}\sqrt{2}+625\left(\sqrt{2}\right)^{2}-2\times 2\times 50\left(\sqrt{6}-\sqrt{2}\right)\times 633^{2}$$

The square of \sqrt{6} is 6.

$$2500+625\times 6-1250\sqrt{6}\sqrt{2}+625\left(\sqrt{2}\right)^{2}-2\times 2\times 50\left(\sqrt{6}-\sqrt{2}\right)\times 633^{2}$$

Multiply 625 and 6 to get 3750.

$$2500+3750-1250\sqrt{6}\sqrt{2}+625\left(\sqrt{2}\right)^{2}-2\times 2\times 50\left(\sqrt{6}-\sqrt{2}\right)\times 633^{2}$$

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

$$2500+3750-1250\sqrt{2}\sqrt{3}\sqrt{2}+625\left(\sqrt{2}\right)^{2}-2\times 2\times 50\left(\sqrt{6}-\sqrt{2}\right)\times 633^{2}$$

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

$$2500+3750-1250\times 2\sqrt{3}+625\left(\sqrt{2}\right)^{2}-2\times 2\times 50\left(\sqrt{6}-\sqrt{2}\right)\times 633^{2}$$

Multiply -1250 and 2 to get -2500.

$$2500+3750-2500\sqrt{3}+625\left(\sqrt{2}\right)^{2}-2\times 2\times 50\left(\sqrt{6}-\sqrt{2}\right)\times 633^{2}$$

The square of \sqrt{2} is 2.

$$2500+3750-2500\sqrt{3}+625\times 2-2\times 2\times 50\left(\sqrt{6}-\sqrt{2}\right)\times 633^{2}$$

Multiply 625 and 2 to get 1250.

$$2500+3750-2500\sqrt{3}+1250-2\times 2\times 50\left(\sqrt{6}-\sqrt{2}\right)\times 633^{2}$$

Add 3750 and 1250 to get 5000.

$$2500+5000-2500\sqrt{3}-2\times 2\times 50\left(\sqrt{6}-\sqrt{2}\right)\times 633^{2}$$

Add 2500 and 5000 to get 7500.

$$7500-2500\sqrt{3}-2\times 2\times 50\left(\sqrt{6}-\sqrt{2}\right)\times 633^{2}$$

Multiply 2 and 2 to get 4.

$$7500-2500\sqrt{3}-4\times 50\left(\sqrt{6}-\sqrt{2}\right)\times 633^{2}$$

Multiply 4 and 50 to get 200.

$$7500-2500\sqrt{3}-200\left(\sqrt{6}-\sqrt{2}\right)\times 633^{2}$$

Calculate 633 to the power of 2 and get 400689.

$$7500-2500\sqrt{3}-200\left(\sqrt{6}-\sqrt{2}\right)\times 400689$$

Multiply 200 and 400689 to get 80137800.

$$7500-2500\sqrt{3}-80137800\left(\sqrt{6}-\sqrt{2}\right)$$

Use the distributive property to multiply -80137800 by \sqrt{6}-\sqrt{2}.

$$7500-2500\sqrt{3}-80137800\sqrt{6}[phone]\sqrt{2}$$

Factor

$$100 {(801378 \sqrt{2} + 75 - 25 \sqrt{3} - 801378 \sqrt{6})} = -82961585.61748604$$