Solve 57/5sqrt{3-3sqrt{2}}

$$\frac{ 57 }{ 5 \sqrt{ 3 } -3 \sqrt{ 2 } }$$

Evaluate

$$\sqrt{3}\left(\sqrt{6}+5\right)\approx 12.902894725$$

Solution Steps

Rationalize the denominator of \frac{57}{5\sqrt{3}-3\sqrt{2}} by multiplying numerator and denominator by 5\sqrt{3}+3\sqrt{2}.

$$\frac{57\left(5\sqrt{3}+3\sqrt{2}\right)}{\left(5\sqrt{3}-3\sqrt{2}\right)\left(5\sqrt{3}+3\sqrt{2}\right)}$$

Consider \left(5\sqrt{3}-3\sqrt{2}\right)\left(5\sqrt{3}+3\sqrt{2}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{57\left(5\sqrt{3}+3\sqrt{2}\right)}{\left(5\sqrt{3}\right)^{2}-\left(-3\sqrt{2}\right)^{2}}$$

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

$$\frac{57\left(5\sqrt{3}+3\sqrt{2}\right)}{5^{2}\left(\sqrt{3}\right)^{2}-\left(-3\sqrt{2}\right)^{2}}$$

Calculate 5 to the power of 2 and get 25.

$$\frac{57\left(5\sqrt{3}+3\sqrt{2}\right)}{25\left(\sqrt{3}\right)^{2}-\left(-3\sqrt{2}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{57\left(5\sqrt{3}+3\sqrt{2}\right)}{25\times 3-\left(-3\sqrt{2}\right)^{2}}$$

Multiply 25 and 3 to get 75.

$$\frac{57\left(5\sqrt{3}+3\sqrt{2}\right)}{75-\left(-3\sqrt{2}\right)^{2}}$$

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

$$\frac{57\left(5\sqrt{3}+3\sqrt{2}\right)}{75-\left(-3\right)^{2}\left(\sqrt{2}\right)^{2}}$$

Calculate -3 to the power of 2 and get 9.

$$\frac{57\left(5\sqrt{3}+3\sqrt{2}\right)}{75-9\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{57\left(5\sqrt{3}+3\sqrt{2}\right)}{75-9\times 2}$$

Multiply 9 and 2 to get 18.

$$\frac{57\left(5\sqrt{3}+3\sqrt{2}\right)}{75-18}$$

Subtract 18 from 75 to get 57.

$$\frac{57\left(5\sqrt{3}+3\sqrt{2}\right)}{57}$$

Cancel out 57 and 57.

$$5\sqrt{3}+3\sqrt{2}$$