Solve frac{6sqrt{3}}{sqrt{3}-1}*frac{sqrt{3+1}}{sqrt{3}+1}

$$\frac{ 6 \sqrt{ 3 } }{ \sqrt{ 3 } -1 } \times \frac{ \sqrt{ 3+1 } }{ \sqrt{ 3 } +1 }$$

Evaluate

$$6\sqrt{3}\approx 10.392304845$$

Solution Steps

Rationalize the denominator of \frac{6\sqrt{3}}{\sqrt{3}-1} by multiplying numerator and denominator by \sqrt{3}+1.

$$\frac{6\sqrt{3}\left(\sqrt{3}+1\right)}{\left(\sqrt{3}-1\right)\left(\sqrt{3}+1\right)}\times \frac{\sqrt{3+1}}{\sqrt{3}+1}$$

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

$$\frac{6\sqrt{3}\left(\sqrt{3}+1\right)}{\left(\sqrt{3}\right)^{2}-1^{2}}\times \frac{\sqrt{3+1}}{\sqrt{3}+1}$$

Square \sqrt{3}. Square 1.

$$\frac{6\sqrt{3}\left(\sqrt{3}+1\right)}{3-1}\times \frac{\sqrt{3+1}}{\sqrt{3}+1}$$

Subtract 1 from 3 to get 2.

$$\frac{6\sqrt{3}\left(\sqrt{3}+1\right)}{2}\times \frac{\sqrt{3+1}}{\sqrt{3}+1}$$

Add 3 and 1 to get 4.

$$\frac{6\sqrt{3}\left(\sqrt{3}+1\right)}{2}\times \frac{\sqrt{4}}{\sqrt{3}+1}$$

Calculate the square root of 4 and get 2.

$$\frac{6\sqrt{3}\left(\sqrt{3}+1\right)}{2}\times \frac{2}{\sqrt{3}+1}$$

Rationalize the denominator of \frac{2}{\sqrt{3}+1} by multiplying numerator and denominator by \sqrt{3}-1.

$$\frac{6\sqrt{3}\left(\sqrt{3}+1\right)}{2}\times \frac{2\left(\sqrt{3}-1\right)}{\left(\sqrt{3}+1\right)\left(\sqrt{3}-1\right)}$$

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

$$\frac{6\sqrt{3}\left(\sqrt{3}+1\right)}{2}\times \frac{2\left(\sqrt{3}-1\right)}{\left(\sqrt{3}\right)^{2}-1^{2}}$$

Square \sqrt{3}. Square 1.

$$\frac{6\sqrt{3}\left(\sqrt{3}+1\right)}{2}\times \frac{2\left(\sqrt{3}-1\right)}{3-1}$$

Subtract 1 from 3 to get 2.

$$\frac{6\sqrt{3}\left(\sqrt{3}+1\right)}{2}\times \frac{2\left(\sqrt{3}-1\right)}{2}$$

Cancel out 2 and 2.

$$\frac{6\sqrt{3}\left(\sqrt{3}+1\right)}{2}\left(\sqrt{3}-1\right)$$

Express \frac{6\sqrt{3}\left(\sqrt{3}+1\right)}{2}\left(\sqrt{3}-1\right) as a single fraction.

$$\frac{6\sqrt{3}\left(\sqrt{3}+1\right)\left(\sqrt{3}-1\right)}{2}$$

Use the distributive property to multiply 6\sqrt{3} by \sqrt{3}+1.

$$\frac{\left(6\left(\sqrt{3}\right)^{2}+6\sqrt{3}\right)\left(\sqrt{3}-1\right)}{2}$$

The square of \sqrt{3} is 3.

$$\frac{\left(6\times 3+6\sqrt{3}\right)\left(\sqrt{3}-1\right)}{2}$$

Multiply 6 and 3 to get 18.

$$\frac{\left(18+6\sqrt{3}\right)\left(\sqrt{3}-1\right)}{2}$$

Apply the distributive property by multiplying each term of 18+6\sqrt{3} by each term of \sqrt{3}-1.

$$\frac{18\sqrt{3}-18+6\left(\sqrt{3}\right)^{2}-6\sqrt{3}}{2}$$

The square of \sqrt{3} is 3.

$$\frac{18\sqrt{3}-18+6\times 3-6\sqrt{3}}{2}$$

Multiply 6 and 3 to get 18.

$$\frac{18\sqrt{3}-18+18-6\sqrt{3}}{2}$$

Add -18 and 18 to get 0.

$$\frac{18\sqrt{3}-6\sqrt{3}}{2}$$

Combine 18\sqrt{3} and -6\sqrt{3} to get 12\sqrt{3}.

$$\frac{12\sqrt{3}}{2}$$

Divide 12\sqrt{3} by 2 to get 6\sqrt{3}.

$$6\sqrt{3}$$