Solve {l}{(47+sqrt{5})^2-2*(sqrt{7}+sqrt{5})*(sqrt{7}-sqrt{5})}{+(sqrt{7}-sqrt{5})^2}

$$\left. \begin{array} { l } { ( 47 + \sqrt { 5 } ) ^ { 2 } - 2 \cdot ( \sqrt { 7 } + \sqrt { 5 } ) \cdot ( \sqrt { 7 } - \sqrt { 5 } ) } \\ { + ( \sqrt { 7 } - \sqrt { 5 } ) ^ { 2 } } \end{array} \right.$$

Evaluate

$$94\sqrt{5}+2222-2\sqrt{35}\approx 2420.358230319$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(47+\sqrt{5}\right)^{2}.

$$2209+94\sqrt{5}+\left(\sqrt{5}\right)^{2}-2\left(\sqrt{7}+\sqrt{5}\right)\left(\sqrt{7}-\sqrt{5}\right)+\left(\sqrt{7}-\sqrt{5}\right)^{2}$$

The square of \sqrt{5} is 5.

$$2209+94\sqrt{5}+5-2\left(\sqrt{7}+\sqrt{5}\right)\left(\sqrt{7}-\sqrt{5}\right)+\left(\sqrt{7}-\sqrt{5}\right)^{2}$$

Add 2209 and 5 to get 2214.

$$2214+94\sqrt{5}-2\left(\sqrt{7}+\sqrt{5}\right)\left(\sqrt{7}-\sqrt{5}\right)+\left(\sqrt{7}-\sqrt{5}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\sqrt{7}-\sqrt{5}\right)^{2}.

$$2214+94\sqrt{5}-2\left(\sqrt{7}+\sqrt{5}\right)\left(\sqrt{7}-\sqrt{5}\right)+\left(\sqrt{7}\right)^{2}-2\sqrt{7}\sqrt{5}+\left(\sqrt{5}\right)^{2}$$

The square of \sqrt{7} is 7.

$$2214+94\sqrt{5}-2\left(\sqrt{7}+\sqrt{5}\right)\left(\sqrt{7}-\sqrt{5}\right)+7-2\sqrt{7}\sqrt{5}+\left(\sqrt{5}\right)^{2}$$

To multiply \sqrt{7} and \sqrt{5}, multiply the numbers under the square root.

$$2214+94\sqrt{5}-2\left(\sqrt{7}+\sqrt{5}\right)\left(\sqrt{7}-\sqrt{5}\right)+7-2\sqrt{35}+\left(\sqrt{5}\right)^{2}$$

The square of \sqrt{5} is 5.

$$2214+94\sqrt{5}-2\left(\sqrt{7}+\sqrt{5}\right)\left(\sqrt{7}-\sqrt{5}\right)+7-2\sqrt{35}+5$$

Add 7 and 5 to get 12.

$$2214+94\sqrt{5}-2\left(\sqrt{7}+\sqrt{5}\right)\left(\sqrt{7}-\sqrt{5}\right)+12-2\sqrt{35}$$

Use the distributive property to multiply -2 by \sqrt{7}+\sqrt{5}.

$$2214+94\sqrt{5}+\left(-2\sqrt{7}-2\sqrt{5}\right)\left(\sqrt{7}-\sqrt{5}\right)+12-2\sqrt{35}$$

Use the distributive property to multiply -2\sqrt{7}-2\sqrt{5} by \sqrt{7}-\sqrt{5} and combine like terms.

$$2214+94\sqrt{5}-2\left(\sqrt{7}\right)^{2}+2\left(\sqrt{5}\right)^{2}+12-2\sqrt{35}$$

The square of \sqrt{7} is 7.

$$2214+94\sqrt{5}-2\times 7+2\left(\sqrt{5}\right)^{2}+12-2\sqrt{35}$$

Multiply -2 and 7 to get -14.

$$2214+94\sqrt{5}-14+2\left(\sqrt{5}\right)^{2}+12-2\sqrt{35}$$

The square of \sqrt{5} is 5.

$$2214+94\sqrt{5}-14+2\times 5+12-2\sqrt{35}$$

Multiply 2 and 5 to get 10.

$$2214+94\sqrt{5}-14+10+12-2\sqrt{35}$$

Add -14 and 10 to get -4.

$$2214+94\sqrt{5}-4+12-2\sqrt{35}$$

Subtract 4 from 2214 to get 2210.

$$2210+94\sqrt{5}+12-2\sqrt{35}$$

Add 2210 and 12 to get 2222.

$$2222+94\sqrt{5}-2\sqrt{35}$$