Solve frac{2(3+5)(3-sqrt{5})}{2+sqrt{5}}=

$$\frac { 2 ( 3 + 5 ) ( 3 - \sqrt { 5 } ) } { 2 + \sqrt { 5 } } =$$

Evaluate

$$16\left(5\sqrt{5}-11\right)\approx 2.8854382$$

Solution Steps

Add 3 and 5 to get 8.

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

Multiply 2 and 8 to get 16.

$$\frac{16\left(3-\sqrt{5}\right)}{2+\sqrt{5}}$$

Rationalize the denominator of \frac{16\left(3-\sqrt{5}\right)}{2+\sqrt{5}} by multiplying numerator and denominator by 2-\sqrt{5}.

$$\frac{16\left(3-\sqrt{5}\right)\left(2-\sqrt{5}\right)}{\left(2+\sqrt{5}\right)\left(2-\sqrt{5}\right)}$$

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}.

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

Square 2. Square \sqrt{5}.

$$\frac{16\left(3-\sqrt{5}\right)\left(2-\sqrt{5}\right)}{4-5}$$

Subtract 5 from 4 to get -1.

$$\frac{16\left(3-\sqrt{5}\right)\left(2-\sqrt{5}\right)}{-1}$$

Anything divided by -1 gives its opposite.

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

Use the distributive property to multiply 16 by 3-\sqrt{5}.

$$-\left(48-16\sqrt{5}\right)\left(2-\sqrt{5}\right)$$

Apply the distributive property by multiplying each term of 48-16\sqrt{5} by each term of 2-\sqrt{5}.

$$-\left(96-48\sqrt{5}-32\sqrt{5}+16\left(\sqrt{5}\right)^{2}\right)$$

Combine -48\sqrt{5} and -32\sqrt{5} to get -80\sqrt{5}.

$$-\left(96-80\sqrt{5}+16\left(\sqrt{5}\right)^{2}\right)$$

The square of \sqrt{5} is 5.

$$-\left(96-80\sqrt{5}+16\times 5\right)$$

Multiply 16 and 5 to get 80.

$$-\left(96-80\sqrt{5}+80\right)$$

Add 96 and 80 to get 176.

$$-\left(176-80\sqrt{5}\right)$$

To find the opposite of 176-80\sqrt{5}, find the opposite of each term.

$$-176-\left(-80\sqrt{5}\right)$$

The opposite of -80\sqrt{5} is 80\sqrt{5}.

$$-176+80\sqrt{5}$$