Solve frac{40sqrt{3}}{1+sqrt{3}}*frac{1-sqrt{3}}{1-sqrt{3}}

$$\frac{ 40 \sqrt{ 3 } }{ 1+ \sqrt{ 3 } } \times \frac{ 1- \sqrt{ 3 } }{ 1- \sqrt{ 3 } }$$

Evaluate

$$60-20\sqrt{3}\approx 25.358983849$$

Solution Steps

Divide 1-\sqrt{3} by 1-\sqrt{3} to get 1.

$$\frac{40\sqrt{3}}{1+\sqrt{3}}\times 1$$

Rationalize the denominator of \frac{40\sqrt{3}}{1+\sqrt{3}} by multiplying numerator and denominator by 1-\sqrt{3}.

$$\frac{40\sqrt{3}\left(1-\sqrt{3}\right)}{\left(1+\sqrt{3}\right)\left(1-\sqrt{3}\right)}\times 1$$

Consider \left(1+\sqrt{3}\right)\left(1-\sqrt{3}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{40\sqrt{3}\left(1-\sqrt{3}\right)}{1^{2}-\left(\sqrt{3}\right)^{2}}\times 1$$

Square 1. Square \sqrt{3}.

$$\frac{40\sqrt{3}\left(1-\sqrt{3}\right)}{1-3}\times 1$$

Subtract 3 from 1 to get -2.

$$\frac{40\sqrt{3}\left(1-\sqrt{3}\right)}{-2}\times 1$$

Express \frac{40\sqrt{3}\left(1-\sqrt{3}\right)}{-2}\times 1 as a single fraction.

$$\frac{40\sqrt{3}\left(1-\sqrt{3}\right)}{-2}$$

Use the distributive property to multiply 40\sqrt{3} by 1-\sqrt{3}.

$$\frac{40\sqrt{3}-40\left(\sqrt{3}\right)^{2}}{-2}$$

The square of \sqrt{3} is 3.

$$\frac{40\sqrt{3}-40\times 3}{-2}$$

Multiply -40 and 3 to get -120.

$$\frac{40\sqrt{3}-120}{-2}$$