Solve (6+2sqrt{5})(1-sqrt{5})^2

$$( 6 + 2 \sqrt { 5 } ) ( 1 - \sqrt { 5 } ) ^ { 2 }$$

Evaluate

$$16$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(1-\sqrt{5}\right)^{2}.

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

The square of \sqrt{5} is 5.

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

Add 1 and 5 to get 6.

$$\left(6+2\sqrt{5}\right)\left(6-2\sqrt{5}\right)$$

Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 6.

$$36-\left(2\sqrt{5}\right)^{2}$$

Expand \left(2\sqrt{5}\right)^{2}.

$$36-2^{2}\left(\sqrt{5}\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$36-4\left(\sqrt{5}\right)^{2}$$

The square of \sqrt{5} is 5.

$$36-4\times 5$$

Multiply 4 and 5 to get 20.

$$36-20$$

Subtract 20 from 36 to get 16.

$$16$$

Factor

$$2^{4}$$