Solve (5)(2sqrt{5}+5sqrt{2})^2-(2sqrt{5}-5sqrt{2})^2

$$( 5 ) ( 2 \sqrt { 5 } + 5 \sqrt { 2 } ) ^ { 2 } - ( 2 \sqrt { 5 } - 5 \sqrt { 2 } ) ^ { 2 }$$

Evaluate

$$120\sqrt{10}+280\approx 659.47331922$$

Solution Steps

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

$$5\left(4\left(\sqrt{5}\right)^{2}+20\sqrt{5}\sqrt{2}+25\left(\sqrt{2}\right)^{2}\right)-\left(2\sqrt{5}-5\sqrt{2}\right)^{2}$$

The square of \sqrt{5} is 5.

$$5\left(4\times 5+20\sqrt{5}\sqrt{2}+25\left(\sqrt{2}\right)^{2}\right)-\left(2\sqrt{5}-5\sqrt{2}\right)^{2}$$

Multiply 4 and 5 to get 20.

$$5\left(20+20\sqrt{5}\sqrt{2}+25\left(\sqrt{2}\right)^{2}\right)-\left(2\sqrt{5}-5\sqrt{2}\right)^{2}$$

To multiply \sqrt{5} and \sqrt{2}, multiply the numbers under the square root.

$$5\left(20+20\sqrt{10}+25\left(\sqrt{2}\right)^{2}\right)-\left(2\sqrt{5}-5\sqrt{2}\right)^{2}$$

The square of \sqrt{2} is 2.

$$5\left(20+20\sqrt{10}+25\times 2\right)-\left(2\sqrt{5}-5\sqrt{2}\right)^{2}$$

Multiply 25 and 2 to get 50.

$$5\left(20+20\sqrt{10}+50\right)-\left(2\sqrt{5}-5\sqrt{2}\right)^{2}$$

Add 20 and 50 to get 70.

$$5\left(70+20\sqrt{10}\right)-\left(2\sqrt{5}-5\sqrt{2}\right)^{2}$$

Use the distributive property to multiply 5 by 70+20\sqrt{10}.

$$350+100\sqrt{10}-\left(2\sqrt{5}-5\sqrt{2}\right)^{2}$$

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

$$350+100\sqrt{10}-\left(4\left(\sqrt{5}\right)^{2}-20\sqrt{5}\sqrt{2}+25\left(\sqrt{2}\right)^{2}\right)$$

The square of \sqrt{5} is 5.

$$350+100\sqrt{10}-\left(4\times 5-20\sqrt{5}\sqrt{2}+25\left(\sqrt{2}\right)^{2}\right)$$

Multiply 4 and 5 to get 20.

$$350+100\sqrt{10}-\left(20-20\sqrt{5}\sqrt{2}+25\left(\sqrt{2}\right)^{2}\right)$$

To multiply \sqrt{5} and \sqrt{2}, multiply the numbers under the square root.

$$350+100\sqrt{10}-\left(20-20\sqrt{10}+25\left(\sqrt{2}\right)^{2}\right)$$

The square of \sqrt{2} is 2.

$$350+100\sqrt{10}-\left(20-20\sqrt{10}+25\times 2\right)$$

Multiply 25 and 2 to get 50.

$$350+100\sqrt{10}-\left(20-20\sqrt{10}+50\right)$$

Add 20 and 50 to get 70.

$$350+100\sqrt{10}-\left(70-20\sqrt{10}\right)$$

To find the opposite of 70-20\sqrt{10}, find the opposite of each term.

$$350+100\sqrt{10}-70+20\sqrt{10}$$

Subtract 70 from 350 to get 280.

$$280+100\sqrt{10}+20\sqrt{10}$$

Combine 100\sqrt{10} and 20\sqrt{10} to get 120\sqrt{10}.

$$280+120\sqrt{10}$$

Expand

$$120 \sqrt{10} + 280 = 659.47331922$$

Solution Steps

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

$$5\left(4\left(\sqrt{5}\right)^{2}+20\sqrt{5}\sqrt{2}+25\left(\sqrt{2}\right)^{2}\right)-\left(2\sqrt{5}-5\sqrt{2}\right)^{2}$$

The square of \sqrt{5} is 5.

$$5\left(4\times 5+20\sqrt{5}\sqrt{2}+25\left(\sqrt{2}\right)^{2}\right)-\left(2\sqrt{5}-5\sqrt{2}\right)^{2}$$

Multiply 4 and 5 to get 20.

$$5\left(20+20\sqrt{5}\sqrt{2}+25\left(\sqrt{2}\right)^{2}\right)-\left(2\sqrt{5}-5\sqrt{2}\right)^{2}$$

To multiply \sqrt{5} and \sqrt{2}, multiply the numbers under the square root.

$$5\left(20+20\sqrt{10}+25\left(\sqrt{2}\right)^{2}\right)-\left(2\sqrt{5}-5\sqrt{2}\right)^{2}$$

The square of \sqrt{2} is 2.

$$5\left(20+20\sqrt{10}+25\times 2\right)-\left(2\sqrt{5}-5\sqrt{2}\right)^{2}$$

Multiply 25 and 2 to get 50.

$$5\left(20+20\sqrt{10}+50\right)-\left(2\sqrt{5}-5\sqrt{2}\right)^{2}$$

Add 20 and 50 to get 70.

$$5\left(70+20\sqrt{10}\right)-\left(2\sqrt{5}-5\sqrt{2}\right)^{2}$$

Use the distributive property to multiply 5 by 70+20\sqrt{10}.

$$350+100\sqrt{10}-\left(2\sqrt{5}-5\sqrt{2}\right)^{2}$$

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

$$350+100\sqrt{10}-\left(4\left(\sqrt{5}\right)^{2}-20\sqrt{5}\sqrt{2}+25\left(\sqrt{2}\right)^{2}\right)$$

The square of \sqrt{5} is 5.

$$350+100\sqrt{10}-\left(4\times 5-20\sqrt{5}\sqrt{2}+25\left(\sqrt{2}\right)^{2}\right)$$

Multiply 4 and 5 to get 20.

$$350+100\sqrt{10}-\left(20-20\sqrt{5}\sqrt{2}+25\left(\sqrt{2}\right)^{2}\right)$$

To multiply \sqrt{5} and \sqrt{2}, multiply the numbers under the square root.

$$350+100\sqrt{10}-\left(20-20\sqrt{10}+25\left(\sqrt{2}\right)^{2}\right)$$

The square of \sqrt{2} is 2.

$$350+100\sqrt{10}-\left(20-20\sqrt{10}+25\times 2\right)$$

Multiply 25 and 2 to get 50.

$$350+100\sqrt{10}-\left(20-20\sqrt{10}+50\right)$$

Add 20 and 50 to get 70.

$$350+100\sqrt{10}-\left(70-20\sqrt{10}\right)$$

To find the opposite of 70-20\sqrt{10}, find the opposite of each term.

$$350+100\sqrt{10}-70+20\sqrt{10}$$

Subtract 70 from 350 to get 280.

$$280+100\sqrt{10}+20\sqrt{10}$$

Combine 100\sqrt{10} and 20\sqrt{10} to get 120\sqrt{10}.

$$280+120\sqrt{10}$$