Solve frac{2sqrt{5}}{sqrt{6}+sqrt{5}}-frac{3sqrt{2}}{sqrt{15}+3sqrt{2}}

$$\frac { 2 \sqrt { 5 } } { \sqrt { 6 } + \sqrt { 5 } } - \frac { 3 \sqrt { 2 } } { \sqrt { 15 } + 3 \sqrt { 2 } }$$

Evaluate

$$3\sqrt{30}-16\approx 0.431676725$$

Solution Steps

Rationalize the denominator of \frac{2\sqrt{5}}{\sqrt{6}+\sqrt{5}} by multiplying numerator and denominator by \sqrt{6}-\sqrt{5}.

$$\frac{2\sqrt{5}\left(\sqrt{6}-\sqrt{5}\right)}{\left(\sqrt{6}+\sqrt{5}\right)\left(\sqrt{6}-\sqrt{5}\right)}-\frac{3\sqrt{2}}{\sqrt{15}+3\sqrt{2}}$$

Consider \left(\sqrt{6}+\sqrt{5}\right)\left(\sqrt{6}-\sqrt{5}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{2\sqrt{5}\left(\sqrt{6}-\sqrt{5}\right)}{\left(\sqrt{6}\right)^{2}-\left(\sqrt{5}\right)^{2}}-\frac{3\sqrt{2}}{\sqrt{15}+3\sqrt{2}}$$

Square \sqrt{6}. Square \sqrt{5}.

$$\frac{2\sqrt{5}\left(\sqrt{6}-\sqrt{5}\right)}{6-5}-\frac{3\sqrt{2}}{\sqrt{15}+3\sqrt{2}}$$

Subtract 5 from 6 to get 1.

$$\frac{2\sqrt{5}\left(\sqrt{6}-\sqrt{5}\right)}{1}-\frac{3\sqrt{2}}{\sqrt{15}+3\sqrt{2}}$$

Anything divided by one gives itself.

$$2\sqrt{5}\left(\sqrt{6}-\sqrt{5}\right)-\frac{3\sqrt{2}}{\sqrt{15}+3\sqrt{2}}$$

Rationalize the denominator of \frac{3\sqrt{2}}{\sqrt{15}+3\sqrt{2}} by multiplying numerator and denominator by \sqrt{15}-3\sqrt{2}.

$$2\sqrt{5}\left(\sqrt{6}-\sqrt{5}\right)-\frac{3\sqrt{2}\left(\sqrt{15}-3\sqrt{2}\right)}{\left(\sqrt{15}+3\sqrt{2}\right)\left(\sqrt{15}-3\sqrt{2}\right)}$$

Consider \left(\sqrt{15}+3\sqrt{2}\right)\left(\sqrt{15}-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}.

$$2\sqrt{5}\left(\sqrt{6}-\sqrt{5}\right)-\frac{3\sqrt{2}\left(\sqrt{15}-3\sqrt{2}\right)}{\left(\sqrt{15}\right)^{2}-\left(3\sqrt{2}\right)^{2}}$$

The square of \sqrt{15} is 15.

$$2\sqrt{5}\left(\sqrt{6}-\sqrt{5}\right)-\frac{3\sqrt{2}\left(\sqrt{15}-3\sqrt{2}\right)}{15-\left(3\sqrt{2}\right)^{2}}$$

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

$$2\sqrt{5}\left(\sqrt{6}-\sqrt{5}\right)-\frac{3\sqrt{2}\left(\sqrt{15}-3\sqrt{2}\right)}{15-3^{2}\left(\sqrt{2}\right)^{2}}$$

Calculate 3 to the power of 2 and get 9.

$$2\sqrt{5}\left(\sqrt{6}-\sqrt{5}\right)-\frac{3\sqrt{2}\left(\sqrt{15}-3\sqrt{2}\right)}{15-9\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$2\sqrt{5}\left(\sqrt{6}-\sqrt{5}\right)-\frac{3\sqrt{2}\left(\sqrt{15}-3\sqrt{2}\right)}{15-9\times 2}$$

Multiply 9 and 2 to get 18.

$$2\sqrt{5}\left(\sqrt{6}-\sqrt{5}\right)-\frac{3\sqrt{2}\left(\sqrt{15}-3\sqrt{2}\right)}{15-18}$$

Subtract 18 from 15 to get -3.

$$2\sqrt{5}\left(\sqrt{6}-\sqrt{5}\right)-\frac{3\sqrt{2}\left(\sqrt{15}-3\sqrt{2}\right)}{-3}$$

Cancel out -3 and -3.

$$2\sqrt{5}\left(\sqrt{6}-\sqrt{5}\right)-\left(-\sqrt{2}\left(\sqrt{15}-3\sqrt{2}\right)\right)$$

The opposite of -\sqrt{2}\left(\sqrt{15}-3\sqrt{2}\right) is \sqrt{2}\left(\sqrt{15}-3\sqrt{2}\right).

$$2\sqrt{5}\left(\sqrt{6}-\sqrt{5}\right)+\sqrt{2}\left(\sqrt{15}-3\sqrt{2}\right)$$

Use the distributive property to multiply 2\sqrt{5} by \sqrt{6}-\sqrt{5}.

$$2\sqrt{5}\sqrt{6}-2\left(\sqrt{5}\right)^{2}+\sqrt{2}\left(\sqrt{15}-3\sqrt{2}\right)$$

To multiply \sqrt{5} and \sqrt{6}, multiply the numbers under the square root.

$$2\sqrt{30}-2\left(\sqrt{5}\right)^{2}+\sqrt{2}\left(\sqrt{15}-3\sqrt{2}\right)$$

The square of \sqrt{5} is 5.

$$2\sqrt{30}-2\times 5+\sqrt{2}\left(\sqrt{15}-3\sqrt{2}\right)$$

Multiply -2 and 5 to get -10.

$$2\sqrt{30}-10+\sqrt{2}\left(\sqrt{15}-3\sqrt{2}\right)$$

Use the distributive property to multiply \sqrt{2} by \sqrt{15}-3\sqrt{2}.

$$2\sqrt{30}-10+\sqrt{2}\sqrt{15}-3\left(\sqrt{2}\right)^{2}$$

To multiply \sqrt{2} and \sqrt{15}, multiply the numbers under the square root.

$$2\sqrt{30}-10+\sqrt{30}-3\left(\sqrt{2}\right)^{2}$$

The square of \sqrt{2} is 2.

$$2\sqrt{30}-10+\sqrt{30}-3\times 2$$

Multiply -3 and 2 to get -6.

$$2\sqrt{30}-10+\sqrt{30}-6$$

Combine 2\sqrt{30} and \sqrt{30} to get 3\sqrt{30}.

$$3\sqrt{30}-10-6$$

Subtract 6 from -10 to get -16.

$$3\sqrt{30}-16$$

Factor

$$3 \sqrt{30} - 16 = 0.431676725$$