Solve left(2/2-sqrt{5/}+1right)left(2/2-sqrt{5}+3right)left(2/2-sqrt{5}+5right)left(frac{2}{2-sqrt{5}}+7right)

$$\left( \frac{ 2 }{ \frac{ 2- \sqrt{ 5 } }{ } } +1 \right) \left( \frac{ 2 }{ 2- \sqrt{ 5 } } +3 \right) \left( \frac{ 2 }{ 2- \sqrt{ 5 } } +5 \right) \left( \frac{ 2 }{ 2- \sqrt{ 5 } } +7 \right)$$

Evaluate

$$209$$

Solution Steps

Divide 2 by \frac{2-\sqrt{5}}{1} by multiplying 2 by the reciprocal of \frac{2-\sqrt{5}}{1}.

$$\left(\frac{2}{2-\sqrt{5}}+1\right)\left(\frac{2}{2-\sqrt{5}}+3\right)\left(\frac{2}{2-\sqrt{5}}+5\right)\left(\frac{2}{2-\sqrt{5}}+7\right)$$

Rationalize the denominator of \frac{2}{2-\sqrt{5}} by multiplying numerator and denominator by 2+\sqrt{5}.

$$\left(\frac{2\left(2+\sqrt{5}\right)}{\left(2-\sqrt{5}\right)\left(2+\sqrt{5}\right)}+1\right)\left(\frac{2}{2-\sqrt{5}}+3\right)\left(\frac{2}{2-\sqrt{5}}+5\right)\left(\frac{2}{2-\sqrt{5}}+7\right)$$

Consider \left(2-\sqrt{5}\right)\left(2+\sqrt{5}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\left(\frac{2\left(2+\sqrt{5}\right)}{2^{2}-\left(\sqrt{5}\right)^{2}}+1\right)\left(\frac{2}{2-\sqrt{5}}+3\right)\left(\frac{2}{2-\sqrt{5}}+5\right)\left(\frac{2}{2-\sqrt{5}}+7\right)$$

Square 2. Square \sqrt{5}.

$$\left(\frac{2\left(2+\sqrt{5}\right)}{4-5}+1\right)\left(\frac{2}{2-\sqrt{5}}+3\right)\left(\frac{2}{2-\sqrt{5}}+5\right)\left(\frac{2}{2-\sqrt{5}}+7\right)$$

Subtract 5 from 4 to get -1.

$$\left(\frac{2\left(2+\sqrt{5}\right)}{-1}+1\right)\left(\frac{2}{2-\sqrt{5}}+3\right)\left(\frac{2}{2-\sqrt{5}}+5\right)\left(\frac{2}{2-\sqrt{5}}+7\right)$$

Anything divided by -1 gives its opposite.

$$\left(-2\left(2+\sqrt{5}\right)+1\right)\left(\frac{2}{2-\sqrt{5}}+3\right)\left(\frac{2}{2-\sqrt{5}}+5\right)\left(\frac{2}{2-\sqrt{5}}+7\right)$$

Rationalize the denominator of \frac{2}{2-\sqrt{5}} by multiplying numerator and denominator by 2+\sqrt{5}.

$$\left(-2\left(2+\sqrt{5}\right)+1\right)\left(\frac{2\left(2+\sqrt{5}\right)}{\left(2-\sqrt{5}\right)\left(2+\sqrt{5}\right)}+3\right)\left(\frac{2}{2-\sqrt{5}}+5\right)\left(\frac{2}{2-\sqrt{5}}+7\right)$$

Consider \left(2-\sqrt{5}\right)\left(2+\sqrt{5}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\left(-2\left(2+\sqrt{5}\right)+1\right)\left(\frac{2\left(2+\sqrt{5}\right)}{2^{2}-\left(\sqrt{5}\right)^{2}}+3\right)\left(\frac{2}{2-\sqrt{5}}+5\right)\left(\frac{2}{2-\sqrt{5}}+7\right)$$

Square 2. Square \sqrt{5}.

$$\left(-2\left(2+\sqrt{5}\right)+1\right)\left(\frac{2\left(2+\sqrt{5}\right)}{4-5}+3\right)\left(\frac{2}{2-\sqrt{5}}+5\right)\left(\frac{2}{2-\sqrt{5}}+7\right)$$

Subtract 5 from 4 to get -1.

$$\left(-2\left(2+\sqrt{5}\right)+1\right)\left(\frac{2\left(2+\sqrt{5}\right)}{-1}+3\right)\left(\frac{2}{2-\sqrt{5}}+5\right)\left(\frac{2}{2-\sqrt{5}}+7\right)$$

Anything divided by -1 gives its opposite.

$$\left(-2\left(2+\sqrt{5}\right)+1\right)\left(-2\left(2+\sqrt{5}\right)+3\right)\left(\frac{2}{2-\sqrt{5}}+5\right)\left(\frac{2}{2-\sqrt{5}}+7\right)$$

Rationalize the denominator of \frac{2}{2-\sqrt{5}} by multiplying numerator and denominator by 2+\sqrt{5}.

$$\left(-2\left(2+\sqrt{5}\right)+1\right)\left(-2\left(2+\sqrt{5}\right)+3\right)\left(\frac{2\left(2+\sqrt{5}\right)}{\left(2-\sqrt{5}\right)\left(2+\sqrt{5}\right)}+5\right)\left(\frac{2}{2-\sqrt{5}}+7\right)$$

Consider \left(2-\sqrt{5}\right)\left(2+\sqrt{5}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\left(-2\left(2+\sqrt{5}\right)+1\right)\left(-2\left(2+\sqrt{5}\right)+3\right)\left(\frac{2\left(2+\sqrt{5}\right)}{2^{2}-\left(\sqrt{5}\right)^{2}}+5\right)\left(\frac{2}{2-\sqrt{5}}+7\right)$$

Square 2. Square \sqrt{5}.

$$\left(-2\left(2+\sqrt{5}\right)+1\right)\left(-2\left(2+\sqrt{5}\right)+3\right)\left(\frac{2\left(2+\sqrt{5}\right)}{4-5}+5\right)\left(\frac{2}{2-\sqrt{5}}+7\right)$$

Subtract 5 from 4 to get -1.

$$\left(-2\left(2+\sqrt{5}\right)+1\right)\left(-2\left(2+\sqrt{5}\right)+3\right)\left(\frac{2\left(2+\sqrt{5}\right)}{-1}+5\right)\left(\frac{2}{2-\sqrt{5}}+7\right)$$

Anything divided by -1 gives its opposite.

$$\left(-2\left(2+\sqrt{5}\right)+1\right)\left(-2\left(2+\sqrt{5}\right)+3\right)\left(-2\left(2+\sqrt{5}\right)+5\right)\left(\frac{2}{2-\sqrt{5}}+7\right)$$

Rationalize the denominator of \frac{2}{2-\sqrt{5}} by multiplying numerator and denominator by 2+\sqrt{5}.

$$\left(-2\left(2+\sqrt{5}\right)+1\right)\left(-2\left(2+\sqrt{5}\right)+3\right)\left(-2\left(2+\sqrt{5}\right)+5\right)\left(\frac{2\left(2+\sqrt{5}\right)}{\left(2-\sqrt{5}\right)\left(2+\sqrt{5}\right)}+7\right)$$

Consider \left(2-\sqrt{5}\right)\left(2+\sqrt{5}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\left(-2\left(2+\sqrt{5}\right)+1\right)\left(-2\left(2+\sqrt{5}\right)+3\right)\left(-2\left(2+\sqrt{5}\right)+5\right)\left(\frac{2\left(2+\sqrt{5}\right)}{2^{2}-\left(\sqrt{5}\right)^{2}}+7\right)$$

Square 2. Square \sqrt{5}.

$$\left(-2\left(2+\sqrt{5}\right)+1\right)\left(-2\left(2+\sqrt{5}\right)+3\right)\left(-2\left(2+\sqrt{5}\right)+5\right)\left(\frac{2\left(2+\sqrt{5}\right)}{4-5}+7\right)$$

Subtract 5 from 4 to get -1.

$$\left(-2\left(2+\sqrt{5}\right)+1\right)\left(-2\left(2+\sqrt{5}\right)+3\right)\left(-2\left(2+\sqrt{5}\right)+5\right)\left(\frac{2\left(2+\sqrt{5}\right)}{-1}+7\right)$$

Anything divided by -1 gives its opposite.

$$\left(-2\left(2+\sqrt{5}\right)+1\right)\left(-2\left(2+\sqrt{5}\right)+3\right)\left(-2\left(2+\sqrt{5}\right)+5\right)\left(-2\left(2+\sqrt{5}\right)+7\right)$$

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

$$\left(-4-2\sqrt{5}+1\right)\left(-2\left(2+\sqrt{5}\right)+3\right)\left(-2\left(2+\sqrt{5}\right)+5\right)\left(-2\left(2+\sqrt{5}\right)+7\right)$$

Add -4 and 1 to get -3.

$$\left(-3-2\sqrt{5}\right)\left(-2\left(2+\sqrt{5}\right)+3\right)\left(-2\left(2+\sqrt{5}\right)+5\right)\left(-2\left(2+\sqrt{5}\right)+7\right)$$

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

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

Add -4 and 3 to get -1.

$$\left(-3-2\sqrt{5}\right)\left(-1-2\sqrt{5}\right)\left(-2\left(2+\sqrt{5}\right)+5\right)\left(-2\left(2+\sqrt{5}\right)+7\right)$$

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

$$\left(-3-2\sqrt{5}\right)\left(-1-2\sqrt{5}\right)\left(-4-2\sqrt{5}+5\right)\left(-2\left(2+\sqrt{5}\right)+7\right)$$

Add -4 and 5 to get 1.

$$\left(-3-2\sqrt{5}\right)\left(-1-2\sqrt{5}\right)\left(1-2\sqrt{5}\right)\left(-2\left(2+\sqrt{5}\right)+7\right)$$

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

$$\left(-3-2\sqrt{5}\right)\left(-1-2\sqrt{5}\right)\left(1-2\sqrt{5}\right)\left(-4-2\sqrt{5}+7\right)$$

Add -4 and 7 to get 3.

$$\left(-3-2\sqrt{5}\right)\left(-1-2\sqrt{5}\right)\left(1-2\sqrt{5}\right)\left(3-2\sqrt{5}\right)$$

Apply the distributive property by multiplying each term of -3-2\sqrt{5} by each term of -1-2\sqrt{5}.

$$\left(3+6\sqrt{5}+2\sqrt{5}+4\left(\sqrt{5}\right)^{2}\right)\left(1-2\sqrt{5}\right)\left(3-2\sqrt{5}\right)$$

Combine 6\sqrt{5} and 2\sqrt{5} to get 8\sqrt{5}.

$$\left(3+8\sqrt{5}+4\left(\sqrt{5}\right)^{2}\right)\left(1-2\sqrt{5}\right)\left(3-2\sqrt{5}\right)$$

The square of \sqrt{5} is 5.

$$\left(3+8\sqrt{5}+4\times 5\right)\left(1-2\sqrt{5}\right)\left(3-2\sqrt{5}\right)$$

Multiply 4 and 5 to get 20.

$$\left(3+8\sqrt{5}+20\right)\left(1-2\sqrt{5}\right)\left(3-2\sqrt{5}\right)$$

Add 3 and 20 to get 23.

$$\left(23+8\sqrt{5}\right)\left(1-2\sqrt{5}\right)\left(3-2\sqrt{5}\right)$$

Apply the distributive property by multiplying each term of 23+8\sqrt{5} by each term of 1-2\sqrt{5}.

$$\left(23-46\sqrt{5}+8\sqrt{5}-16\left(\sqrt{5}\right)^{2}\right)\left(3-2\sqrt{5}\right)$$

Combine -46\sqrt{5} and 8\sqrt{5} to get -38\sqrt{5}.

$$\left(23-38\sqrt{5}-16\left(\sqrt{5}\right)^{2}\right)\left(3-2\sqrt{5}\right)$$

The square of \sqrt{5} is 5.

$$\left(23-38\sqrt{5}-16\times 5\right)\left(3-2\sqrt{5}\right)$$

Multiply -16 and 5 to get -80.

$$\left(23-38\sqrt{5}-80\right)\left(3-2\sqrt{5}\right)$$

Subtract 80 from 23 to get -57.

$$\left(-57-38\sqrt{5}\right)\left(3-2\sqrt{5}\right)$$

Apply the distributive property by multiplying each term of -57-38\sqrt{5} by each term of 3-2\sqrt{5}.

$$-171+114\sqrt{5}-114\sqrt{5}+76\left(\sqrt{5}\right)^{2}$$

Combine 114\sqrt{5} and -114\sqrt{5} to get 0.

$$-171+76\left(\sqrt{5}\right)^{2}$$

The square of \sqrt{5} is 5.

$$-171+76\times 5$$

Multiply 76 and 5 to get 380.

$$-171+380$$

Add -171 and 380 to get 209.

$$209$$

Factor

$$11\times 19$$