Solve frac{7+3sqrt{5}}{2}*frac{7-3sqrt{5}}{2}

$$\frac { 7 + 3 \sqrt { 5 } } { 2 } \times \frac { 7 - 3 \sqrt { 5 } } { 2 }$$

Evaluate

$$1$$

Solution Steps

Multiply \frac{7+3\sqrt{5}}{2} times \frac{7-3\sqrt{5}}{2} by multiplying numerator times numerator and denominator times denominator.

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

Consider \left(7+3\sqrt{5}\right)\left(7-3\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}.

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

Calculate 7 to the power of 2 and get 49.

$$\frac{49-\left(3\sqrt{5}\right)^{2}}{2\times 2}$$

Expand \left(3\sqrt{5}\right)^{2}.

$$\frac{49-3^{2}\left(\sqrt{5}\right)^{2}}{2\times 2}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{49-9\left(\sqrt{5}\right)^{2}}{2\times 2}$$

The square of \sqrt{5} is 5.

$$\frac{49-9\times 5}{2\times 2}$$

Multiply 9 and 5 to get 45.

$$\frac{49-45}{2\times 2}$$

Subtract 45 from 49 to get 4.

$$\frac{4}{2\times 2}$$

Multiply 2 and 2 to get 4.

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

Divide 4 by 4 to get 1.

$$1$$

Factor

$$1$$