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

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

Evaluate

$$\frac{47\sqrt{5}-56\sqrt{2}}{37}\approx 0.699979336$$

Solution Steps

Rationalize the denominator of \frac{\left(\sqrt{5}-\sqrt{2}\right)\left(3\sqrt{5}+\sqrt{2}\right)}{3\sqrt{5}+2\sqrt{2}} by multiplying numerator and denominator by 3\sqrt{5}-2\sqrt{2}.

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

Consider \left(3\sqrt{5}+2\sqrt{2}\right)\left(3\sqrt{5}-2\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}.

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

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

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

Calculate 3 to the power of 2 and get 9.

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

The square of \sqrt{5} is 5.

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

Multiply 9 and 5 to get 45.

$$\frac{\left(\sqrt{5}-\sqrt{2}\right)\left(3\sqrt{5}+\sqrt{2}\right)\left(3\sqrt{5}-2\sqrt{2}\right)}{45-\left(2\sqrt{2}\right)^{2}}$$

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

$$\frac{\left(\sqrt{5}-\sqrt{2}\right)\left(3\sqrt{5}+\sqrt{2}\right)\left(3\sqrt{5}-2\sqrt{2}\right)}{45-2^{2}\left(\sqrt{2}\right)^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{\left(\sqrt{5}-\sqrt{2}\right)\left(3\sqrt{5}+\sqrt{2}\right)\left(3\sqrt{5}-2\sqrt{2}\right)}{45-4\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{\left(\sqrt{5}-\sqrt{2}\right)\left(3\sqrt{5}+\sqrt{2}\right)\left(3\sqrt{5}-2\sqrt{2}\right)}{45-4\times 2}$$

Multiply 4 and 2 to get 8.

$$\frac{\left(\sqrt{5}-\sqrt{2}\right)\left(3\sqrt{5}+\sqrt{2}\right)\left(3\sqrt{5}-2\sqrt{2}\right)}{45-8}$$

Subtract 8 from 45 to get 37.

$$\frac{\left(\sqrt{5}-\sqrt{2}\right)\left(3\sqrt{5}+\sqrt{2}\right)\left(3\sqrt{5}-2\sqrt{2}\right)}{37}$$

Apply the distributive property by multiplying each term of \sqrt{5}-\sqrt{2} by each term of 3\sqrt{5}+\sqrt{2}.

$$\frac{\left(3\left(\sqrt{5}\right)^{2}+\sqrt{5}\sqrt{2}-3\sqrt{2}\sqrt{5}-\left(\sqrt{2}\right)^{2}\right)\left(3\sqrt{5}-2\sqrt{2}\right)}{37}$$

The square of \sqrt{5} is 5.

$$\frac{\left(3\times 5+\sqrt{5}\sqrt{2}-3\sqrt{2}\sqrt{5}-\left(\sqrt{2}\right)^{2}\right)\left(3\sqrt{5}-2\sqrt{2}\right)}{37}$$

Multiply 3 and 5 to get 15.

$$\frac{\left(15+\sqrt{5}\sqrt{2}-3\sqrt{2}\sqrt{5}-\left(\sqrt{2}\right)^{2}\right)\left(3\sqrt{5}-2\sqrt{2}\right)}{37}$$

To multiply \sqrt{5} and \sqrt{2}, multiply the numbers under the square root.

$$\frac{\left(15+\sqrt{10}-3\sqrt{2}\sqrt{5}-\left(\sqrt{2}\right)^{2}\right)\left(3\sqrt{5}-2\sqrt{2}\right)}{37}$$

To multiply \sqrt{2} and \sqrt{5}, multiply the numbers under the square root.

$$\frac{\left(15+\sqrt{10}-3\sqrt{10}-\left(\sqrt{2}\right)^{2}\right)\left(3\sqrt{5}-2\sqrt{2}\right)}{37}$$

Combine \sqrt{10} and -3\sqrt{10} to get -2\sqrt{10}.

$$\frac{\left(15-2\sqrt{10}-\left(\sqrt{2}\right)^{2}\right)\left(3\sqrt{5}-2\sqrt{2}\right)}{37}$$

The square of \sqrt{2} is 2.

$$\frac{\left(15-2\sqrt{10}-2\right)\left(3\sqrt{5}-2\sqrt{2}\right)}{37}$$

Subtract 2 from 15 to get 13.

$$\frac{\left(13-2\sqrt{10}\right)\left(3\sqrt{5}-2\sqrt{2}\right)}{37}$$

Apply the distributive property by multiplying each term of 13-2\sqrt{10} by each term of 3\sqrt{5}-2\sqrt{2}.

$$\frac{39\sqrt{5}-26\sqrt{2}-6\sqrt{10}\sqrt{5}+4\sqrt{2}\sqrt{10}}{37}$$

Factor 10=5\times 2. Rewrite the square root of the product \sqrt{5\times 2} as the product of square roots \sqrt{5}\sqrt{2}.

$$\frac{39\sqrt{5}-26\sqrt{2}-6\sqrt{5}\sqrt{2}\sqrt{5}+4\sqrt{2}\sqrt{10}}{37}$$

Multiply \sqrt{5} and \sqrt{5} to get 5.

$$\frac{39\sqrt{5}-26\sqrt{2}-6\times 5\sqrt{2}+4\sqrt{2}\sqrt{10}}{37}$$

Multiply -6 and 5 to get -30.

$$\frac{39\sqrt{5}-26\sqrt{2}-30\sqrt{2}+4\sqrt{2}\sqrt{10}}{37}$$

Combine -26\sqrt{2} and -30\sqrt{2} to get -56\sqrt{2}.

$$\frac{39\sqrt{5}-56\sqrt{2}+4\sqrt{2}\sqrt{10}}{37}$$

Factor 10=2\times 5. Rewrite the square root of the product \sqrt{2\times 5} as the product of square roots \sqrt{2}\sqrt{5}.

$$\frac{39\sqrt{5}-56\sqrt{2}+4\sqrt{2}\sqrt{2}\sqrt{5}}{37}$$

Multiply \sqrt{2} and \sqrt{2} to get 2.

$$\frac{39\sqrt{5}-56\sqrt{2}+4\times 2\sqrt{5}}{37}$$

Multiply 4 and 2 to get 8.

$$\frac{39\sqrt{5}-56\sqrt{2}+8\sqrt{5}}{37}$$

Combine 39\sqrt{5} and 8\sqrt{5} to get 47\sqrt{5}.

$$\frac{47\sqrt{5}-56\sqrt{2}}{37}$$