Solve frac{5sqrt{3}+4sqrt{5}}{5+3sqrt{5}}

$$\frac { 5 \sqrt { 3 } + 4 \sqrt { 5 } } { 5 + 3 \sqrt { 5 } }$$

Evaluate

$$\frac{3\sqrt{15}}{4}-\frac{5\sqrt{3}}{4}+3-\sqrt{5}\approx 1.503606023$$

Solution Steps

Rationalize the denominator of \frac{5\sqrt{3}+4\sqrt{5}}{5+3\sqrt{5}} by multiplying numerator and denominator by 5-3\sqrt{5}.

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

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

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

Calculate 5 to the power of 2 and get 25.

$$\frac{\left(5\sqrt{3}+4\sqrt{5}\right)\left(5-3\sqrt{5}\right)}{25-\left(3\sqrt{5}\right)^{2}}$$

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

$$\frac{\left(5\sqrt{3}+4\sqrt{5}\right)\left(5-3\sqrt{5}\right)}{25-3^{2}\left(\sqrt{5}\right)^{2}}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{\left(5\sqrt{3}+4\sqrt{5}\right)\left(5-3\sqrt{5}\right)}{25-9\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{\left(5\sqrt{3}+4\sqrt{5}\right)\left(5-3\sqrt{5}\right)}{25-9\times 5}$$

Multiply 9 and 5 to get 45.

$$\frac{\left(5\sqrt{3}+4\sqrt{5}\right)\left(5-3\sqrt{5}\right)}{25-45}$$

Subtract 45 from 25 to get -20.

$$\frac{\left(5\sqrt{3}+4\sqrt{5}\right)\left(5-3\sqrt{5}\right)}{-20}$$

Apply the distributive property by multiplying each term of 5\sqrt{3}+4\sqrt{5} by each term of 5-3\sqrt{5}.

$$\frac{25\sqrt{3}-15\sqrt{3}\sqrt{5}+20\sqrt{5}-12\left(\sqrt{5}\right)^{2}}{-20}$$

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

$$\frac{25\sqrt{3}-15\sqrt{15}+20\sqrt{5}-12\left(\sqrt{5}\right)^{2}}{-20}$$

The square of \sqrt{5} is 5.

$$\frac{25\sqrt{3}-15\sqrt{15}+20\sqrt{5}-12\times 5}{-20}$$

Multiply -12 and 5 to get -60.

$$\frac{25\sqrt{3}-15\sqrt{15}+20\sqrt{5}-60}{-20}$$

Factor

$$\frac{3 \sqrt{15} + 12 - 4 \sqrt{5} - 5 \sqrt{3}}{4} = 1.5036060226946768$$