Solve frac{3+sqrt{5}}{5}*frac{3-sqrt{5}}{5}

$$\frac { 3 + \sqrt { 5 } } { 5 } \times \frac { 3 - \sqrt { 5 } } { 5 }$$

Evaluate

$$\frac{4}{25}=0.16$$

Solution Steps

Multiply \frac{3+\sqrt{5}}{5} times \frac{3-\sqrt{5}}{5} by multiplying numerator times numerator and denominator times denominator.

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

Consider \left(3+\sqrt{5}\right)\left(3-\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}.

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

Calculate 3 to the power of 2 and get 9.

$$\frac{9-\left(\sqrt{5}\right)^{2}}{5\times 5}$$

The square of \sqrt{5} is 5.

$$\frac{9-5}{5\times 5}$$

Subtract 5 from 9 to get 4.

$$\frac{4}{5\times 5}$$

Multiply 5 and 5 to get 25.

$$\frac{4}{25}$$

Factor

$$\frac{2 ^ {2}}{5 ^ {2}} = 0.16$$