Solve frac{{left(3sqrt{3}-sqrt{57}right)}^2}{14-3sqrt{19}}

$$\frac{ { \left(3 \sqrt{ 3 } - \sqrt{ 57 } \right) }^{ 2 } }{ 14-3 \sqrt{ 19 } }$$

Evaluate

$$6$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(3\sqrt{3}-\sqrt{57}\right)^{2}.

$$\frac{9\left(\sqrt{3}\right)^{2}-6\sqrt{3}\sqrt{57}+\left(\sqrt{57}\right)^{2}}{14-3\sqrt{19}}$$

The square of \sqrt{3} is 3.

$$\frac{9\times 3-6\sqrt{3}\sqrt{57}+\left(\sqrt{57}\right)^{2}}{14-3\sqrt{19}}$$

Multiply 9 and 3 to get 27.

$$\frac{27-6\sqrt{3}\sqrt{57}+\left(\sqrt{57}\right)^{2}}{14-3\sqrt{19}}$$

Factor 57=3\times 19. Rewrite the square root of the product \sqrt{3\times 19} as the product of square roots \sqrt{3}\sqrt{19}.

$$\frac{27-6\sqrt{3}\sqrt{3}\sqrt{19}+\left(\sqrt{57}\right)^{2}}{14-3\sqrt{19}}$$

Multiply \sqrt{3} and \sqrt{3} to get 3.

$$\frac{27-6\times 3\sqrt{19}+\left(\sqrt{57}\right)^{2}}{14-3\sqrt{19}}$$

Multiply -6 and 3 to get -18.

$$\frac{27-18\sqrt{19}+\left(\sqrt{57}\right)^{2}}{14-3\sqrt{19}}$$

The square of \sqrt{57} is 57.

$$\frac{27-18\sqrt{19}+57}{14-3\sqrt{19}}$$

Add 27 and 57 to get 84.

$$\frac{84-18\sqrt{19}}{14-3\sqrt{19}}$$

Rationalize the denominator of \frac{84-18\sqrt{19}}{14-3\sqrt{19}} by multiplying numerator and denominator by 14+3\sqrt{19}.

$$\frac{\left(84-18\sqrt{19}\right)\left(14+3\sqrt{19}\right)}{\left(14-3\sqrt{19}\right)\left(14+3\sqrt{19}\right)}$$

Consider \left(14-3\sqrt{19}\right)\left(14+3\sqrt{19}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(84-18\sqrt{19}\right)\left(14+3\sqrt{19}\right)}{14^{2}-\left(-3\sqrt{19}\right)^{2}}$$

Calculate 14 to the power of 2 and get 196.

$$\frac{\left(84-18\sqrt{19}\right)\left(14+3\sqrt{19}\right)}{196-\left(-3\sqrt{19}\right)^{2}}$$

Expand \left(-3\sqrt{19}\right)^{2}.

$$\frac{\left(84-18\sqrt{19}\right)\left(14+3\sqrt{19}\right)}{196-\left(-3\right)^{2}\left(\sqrt{19}\right)^{2}}$$

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

$$\frac{\left(84-18\sqrt{19}\right)\left(14+3\sqrt{19}\right)}{196-9\left(\sqrt{19}\right)^{2}}$$

The square of \sqrt{19} is 19.

$$\frac{\left(84-18\sqrt{19}\right)\left(14+3\sqrt{19}\right)}{196-9\times 19}$$

Multiply 9 and 19 to get 171.

$$\frac{\left(84-18\sqrt{19}\right)\left(14+3\sqrt{19}\right)}{196-171}$$

Subtract 171 from 196 to get 25.

$$\frac{\left(84-18\sqrt{19}\right)\left(14+3\sqrt{19}\right)}{25}$$

Use the distributive property to multiply 84-18\sqrt{19} by 14+3\sqrt{19} and combine like terms.

$$\frac{1176-54\left(\sqrt{19}\right)^{2}}{25}$$

The square of \sqrt{19} is 19.

$$\frac{1176-54\times 19}{25}$$

Multiply -54 and 19 to get -1026.

$$\frac{1176-1026}{25}$$

Subtract 1026 from 1176 to get 150.

$$\frac{150}{25}$$

Divide 150 by 25 to get 6.

$$6$$

Factor

$$2\times 3$$