Solve frac{6sqrt{3}-3sqrt{5}}{3sqrt{5}-5sqrt{3}}

$$\frac { 6 \sqrt { 3 } - 3 \sqrt { 5 } } { 3 \sqrt { 5 } - 5 \sqrt { 3 } }$$

Evaluate

$$-\frac{\sqrt{15}}{10}-\frac{3}{2}\approx -1.887298335$$

Solution Steps

Rationalize the denominator of \frac{6\sqrt{3}-3\sqrt{5}}{3\sqrt{5}-5\sqrt{3}} by multiplying numerator and denominator by 3\sqrt{5}+5\sqrt{3}.

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

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

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

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

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

Calculate 3 to the power of 2 and get 9.

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

The square of \sqrt{5} is 5.

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

Multiply 9 and 5 to get 45.

$$\frac{\left(6\sqrt{3}-3\sqrt{5}\right)\left(3\sqrt{5}+5\sqrt{3}\right)}{45-\left(-5\sqrt{3}\right)^{2}}$$

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

$$\frac{\left(6\sqrt{3}-3\sqrt{5}\right)\left(3\sqrt{5}+5\sqrt{3}\right)}{45-\left(-5\right)^{2}\left(\sqrt{3}\right)^{2}}$$

Calculate -5 to the power of 2 and get 25.

$$\frac{\left(6\sqrt{3}-3\sqrt{5}\right)\left(3\sqrt{5}+5\sqrt{3}\right)}{45-25\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{\left(6\sqrt{3}-3\sqrt{5}\right)\left(3\sqrt{5}+5\sqrt{3}\right)}{45-25\times 3}$$

Multiply 25 and 3 to get 75.

$$\frac{\left(6\sqrt{3}-3\sqrt{5}\right)\left(3\sqrt{5}+5\sqrt{3}\right)}{45-75}$$

Subtract 75 from 45 to get -30.

$$\frac{\left(6\sqrt{3}-3\sqrt{5}\right)\left(3\sqrt{5}+5\sqrt{3}\right)}{-30}$$

Apply the distributive property by multiplying each term of 6\sqrt{3}-3\sqrt{5} by each term of 3\sqrt{5}+5\sqrt{3}.

$$\frac{18\sqrt{3}\sqrt{5}+30\left(\sqrt{3}\right)^{2}-9\left(\sqrt{5}\right)^{2}-15\sqrt{3}\sqrt{5}}{-30}$$

To multiply \sqrt{3} and \sqrt{5}, multiply the numbers under the square root.

$$\frac{18\sqrt{15}+30\left(\sqrt{3}\right)^{2}-9\left(\sqrt{5}\right)^{2}-15\sqrt{3}\sqrt{5}}{-30}$$

The square of \sqrt{3} is 3.

$$\frac{18\sqrt{15}+30\times 3-9\left(\sqrt{5}\right)^{2}-15\sqrt{3}\sqrt{5}}{-30}$$

Multiply 30 and 3 to get 90.

$$\frac{18\sqrt{15}+90-9\left(\sqrt{5}\right)^{2}-15\sqrt{3}\sqrt{5}}{-30}$$

The square of \sqrt{5} is 5.

$$\frac{18\sqrt{15}+90-9\times 5-15\sqrt{3}\sqrt{5}}{-30}$$

Multiply -9 and 5 to get -45.

$$\frac{18\sqrt{15}+90-45-15\sqrt{3}\sqrt{5}}{-30}$$

Subtract 45 from 90 to get 45.

$$\frac{18\sqrt{15}+45-15\sqrt{3}\sqrt{5}}{-30}$$

To multiply \sqrt{3} and \sqrt{5}, multiply the numbers under the square root.

$$\frac{18\sqrt{15}+45-15\sqrt{15}}{-30}$$

Combine 18\sqrt{15} and -15\sqrt{15} to get 3\sqrt{15}.

$$\frac{3\sqrt{15}+45}{-30}$$

Factor

$$\frac{-\sqrt{15} - 15}{10} = -1.8872983346207417$$