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

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

Evaluate

$$5\sqrt{3}+9\approx 17.660254038$$

Solution Steps

Rationalize the denominator of \frac{3\sqrt{5}+\sqrt{15}}{2\sqrt{5}-\sqrt{15}} by multiplying numerator and denominator by 2\sqrt{5}+\sqrt{15}.

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

Consider \left(2\sqrt{5}-\sqrt{15}\right)\left(2\sqrt{5}+\sqrt{15}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

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

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

Calculate 2 to the power of 2 and get 4.

$$\frac{\left(3\sqrt{5}+\sqrt{15}\right)\left(2\sqrt{5}+\sqrt{15}\right)}{4\left(\sqrt{5}\right)^{2}-\left(\sqrt{15}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{\left(3\sqrt{5}+\sqrt{15}\right)\left(2\sqrt{5}+\sqrt{15}\right)}{4\times 5-\left(\sqrt{15}\right)^{2}}$$

Multiply 4 and 5 to get 20.

$$\frac{\left(3\sqrt{5}+\sqrt{15}\right)\left(2\sqrt{5}+\sqrt{15}\right)}{20-\left(\sqrt{15}\right)^{2}}$$

The square of \sqrt{15} is 15.

$$\frac{\left(3\sqrt{5}+\sqrt{15}\right)\left(2\sqrt{5}+\sqrt{15}\right)}{20-15}$$

Subtract 15 from 20 to get 5.

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

Apply the distributive property by multiplying each term of 3\sqrt{5}+\sqrt{15} by each term of 2\sqrt{5}+\sqrt{15}.

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

The square of \sqrt{5} is 5.

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

Multiply 6 and 5 to get 30.

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

Factor 15=5\times 3. Rewrite the square root of the product \sqrt{5\times 3} as the product of square roots \sqrt{5}\sqrt{3}.

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

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

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

Multiply 3 and 5 to get 15.

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

Factor 15=5\times 3. Rewrite the square root of the product \sqrt{5\times 3} as the product of square roots \sqrt{5}\sqrt{3}.

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

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

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

Multiply 2 and 5 to get 10.

$$\frac{30+15\sqrt{3}+10\sqrt{3}+\left(\sqrt{15}\right)^{2}}{5}$$

Combine 15\sqrt{3} and 10\sqrt{3} to get 25\sqrt{3}.

$$\frac{30+25\sqrt{3}+\left(\sqrt{15}\right)^{2}}{5}$$

The square of \sqrt{15} is 15.

$$\frac{30+25\sqrt{3}+15}{5}$$

Add 30 and 15 to get 45.

$$\frac{45+25\sqrt{3}}{5}$$

Divide each term of 45+25\sqrt{3} by 5 to get 9+5\sqrt{3}.

$$9+5\sqrt{3}$$