Solve frac{(sqrt{5}-sqrt{2})(sqrt{5}+sqrt{2})}{3(3-sqrt{3})^2}

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

Evaluate

$$\frac{\sqrt{3}+2}{6}\approx 0.622008468$$

Solution Steps

Consider \left(\sqrt{5}-\sqrt{2}\right)\left(\sqrt{5}+\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}.

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

The square of \sqrt{5} is 5.

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

The square of \sqrt{2} is 2.

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

Subtract 2 from 5 to get 3.

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

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

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

The square of \sqrt{3} is 3.

$$\frac{3}{3\left(9-6\sqrt{3}+3\right)}$$

Add 9 and 3 to get 12.

$$\frac{3}{3\left(12-6\sqrt{3}\right)}$$

Use the distributive property to multiply 3 by 12-6\sqrt{3}.

$$\frac{3}{36-18\sqrt{3}}$$

Rationalize the denominator of \frac{3}{36-18\sqrt{3}} by multiplying numerator and denominator by 36+18\sqrt{3}.

$$\frac{3\left(36+18\sqrt{3}\right)}{\left(36-18\sqrt{3}\right)\left(36+18\sqrt{3}\right)}$$

Consider \left(36-18\sqrt{3}\right)\left(36+18\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}.

$$\frac{3\left(36+18\sqrt{3}\right)}{36^{2}-\left(-18\sqrt{3}\right)^{2}}$$

Calculate 36 to the power of 2 and get 1296.

$$\frac{3\left(36+18\sqrt{3}\right)}{1296-\left(-18\sqrt{3}\right)^{2}}$$

Expand \left(-18\sqrt{3}\right)^{2}.

$$\frac{3\left(36+18\sqrt{3}\right)}{1296-\left(-18\right)^{2}\left(\sqrt{3}\right)^{2}}$$

Calculate -18 to the power of 2 and get 324.

$$\frac{3\left(36+18\sqrt{3}\right)}{1296-324\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{3\left(36+18\sqrt{3}\right)}{1296-324\times 3}$$

Multiply 324 and 3 to get 972.

$$\frac{3\left(36+18\sqrt{3}\right)}{1296-972}$$

Subtract 972 from 1296 to get 324.

$$\frac{3\left(36+18\sqrt{3}\right)}{324}$$

Divide 3\left(36+18\sqrt{3}\right) by 324 to get \frac{1}{108}\left(36+18\sqrt{3}\right).

$$\frac{1}{108}\left(36+18\sqrt{3}\right)$$

Use the distributive property to multiply \frac{1}{108} by 36+18\sqrt{3}.

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