Solve frac{sqrt{3}}{-sqrt{15}}-frac{sqrt{5}}{4+sqrt{15}}+18/sqrt{5+sqrt{3}}

$$\frac { \sqrt { 3 } } { - \sqrt { 15 } } - \frac { \sqrt { 5 } } { 4 + \sqrt { 15 } } + \frac { 18 } { \sqrt { 5 } + \sqrt { 3 } }$$

Evaluate

$$\frac{24\sqrt{5}}{5}-4\sqrt{3}\approx 3.804923062$$

Short Solution Steps

Rationalize the denominator of \frac{\sqrt{5}}{4+\sqrt{15}} by multiplying numerator and denominator by 4-\sqrt{15}.

$$\frac{\sqrt{3}}{-\sqrt{15}}-\frac{\sqrt{5}\left(4-\sqrt{15}\right)}{\left(4+\sqrt{15}\right)\left(4-\sqrt{15}\right)}+\frac{18}{\sqrt{5}+\sqrt{3}}$$

Consider \left(4+\sqrt{15}\right)\left(4-\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}.

$$\frac{\sqrt{3}}{-\sqrt{15}}-\frac{\sqrt{5}\left(4-\sqrt{15}\right)}{4^{2}-\left(\sqrt{15}\right)^{2}}+\frac{18}{\sqrt{5}+\sqrt{3}}$$

Square 4. Square \sqrt{15}.

$$\frac{\sqrt{3}}{-\sqrt{15}}-\frac{\sqrt{5}\left(4-\sqrt{15}\right)}{16-15}+\frac{18}{\sqrt{5}+\sqrt{3}}$$

Subtract 15 from 16 to get 1.

$$\frac{\sqrt{3}}{-\sqrt{15}}-\frac{\sqrt{5}\left(4-\sqrt{15}\right)}{1}+\frac{18}{\sqrt{5}+\sqrt{3}}$$

Anything divided by one gives itself.

$$\frac{\sqrt{3}}{-\sqrt{15}}-\sqrt{5}\left(4-\sqrt{15}\right)+\frac{18}{\sqrt{5}+\sqrt{3}}$$

Rationalize the denominator of \frac{18}{\sqrt{5}+\sqrt{3}} by multiplying numerator and denominator by \sqrt{5}-\sqrt{3}.

$$\frac{\sqrt{3}}{-\sqrt{15}}-\sqrt{5}\left(4-\sqrt{15}\right)+\frac{18\left(\sqrt{5}-\sqrt{3}\right)}{\left(\sqrt{5}+\sqrt{3}\right)\left(\sqrt{5}-\sqrt{3}\right)}$$

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

$$\frac{\sqrt{3}}{-\sqrt{15}}-\sqrt{5}\left(4-\sqrt{15}\right)+\frac{18\left(\sqrt{5}-\sqrt{3}\right)}{\left(\sqrt{5}\right)^{2}-\left(\sqrt{3}\right)^{2}}$$

Square \sqrt{5}. Square \sqrt{3}.

$$\frac{\sqrt{3}}{-\sqrt{15}}-\sqrt{5}\left(4-\sqrt{15}\right)+\frac{18\left(\sqrt{5}-\sqrt{3}\right)}{5-3}$$

Subtract 3 from 5 to get 2.

$$\frac{\sqrt{3}}{-\sqrt{15}}-\sqrt{5}\left(4-\sqrt{15}\right)+\frac{18\left(\sqrt{5}-\sqrt{3}\right)}{2}$$

Divide 18\left(\sqrt{5}-\sqrt{3}\right) by 2 to get 9\left(\sqrt{5}-\sqrt{3}\right).

$$\frac{\sqrt{3}}{-\sqrt{15}}-\sqrt{5}\left(4-\sqrt{15}\right)+9\left(\sqrt{5}-\sqrt{3}\right)$$

Use the distributive property to multiply \sqrt{5} by 4-\sqrt{15}.

$$\frac{\sqrt{3}}{-\sqrt{15}}-\left(4\sqrt{5}-\sqrt{5}\sqrt{15}\right)+9\left(\sqrt{5}-\sqrt{3}\right)$$

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}.

$$\frac{\sqrt{3}}{-\sqrt{15}}-\left(4\sqrt{5}-\sqrt{5}\sqrt{5}\sqrt{3}\right)+9\left(\sqrt{5}-\sqrt{3}\right)$$

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

$$\frac{\sqrt{3}}{-\sqrt{15}}-\left(4\sqrt{5}-5\sqrt{3}\right)+9\left(\sqrt{5}-\sqrt{3}\right)$$

To find the opposite of 4\sqrt{5}-5\sqrt{3}, find the opposite of each term.

$$\frac{\sqrt{3}}{-\sqrt{15}}-4\sqrt{5}-\left(-5\sqrt{3}\right)+9\left(\sqrt{5}-\sqrt{3}\right)$$

The opposite of -5\sqrt{3} is 5\sqrt{3}.

$$\frac{\sqrt{3}}{-\sqrt{15}}-4\sqrt{5}+5\sqrt{3}+9\left(\sqrt{5}-\sqrt{3}\right)$$

Use the distributive property to multiply 9 by \sqrt{5}-\sqrt{3}.

$$\frac{\sqrt{3}}{-\sqrt{15}}-4\sqrt{5}+5\sqrt{3}+9\sqrt{5}-9\sqrt{3}$$

Combine -4\sqrt{5} and 9\sqrt{5} to get 5\sqrt{5}.

$$\frac{\sqrt{3}}{-\sqrt{15}}+5\sqrt{5}+5\sqrt{3}-9\sqrt{3}$$

Combine 5\sqrt{3} and -9\sqrt{3} to get -4\sqrt{3}.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 5\sqrt{5}-4\sqrt{3} times \frac{-\sqrt{15}}{-\sqrt{15}}.

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

Since \frac{\sqrt{3}}{-\sqrt{15}} and \frac{\left(5\sqrt{5}-4\sqrt{3}\right)\left(-\sqrt{15}\right)}{-\sqrt{15}} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \sqrt{3}+\left(5\sqrt{5}-4\sqrt{3}\right)\left(-\sqrt{15}\right).

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

Do the calculations in \sqrt{3}-25\sqrt{3}+12\sqrt{5}.

$$\frac{-24\sqrt{3}+12\sqrt{5}}{-\sqrt{15}}$$

Rationalize the denominator of \frac{-24\sqrt{3}+12\sqrt{5}}{-\sqrt{15}} by multiplying numerator and denominator by \sqrt{15}.

$$\frac{\left(-24\sqrt{3}+12\sqrt{5}\right)\sqrt{15}}{-\left(\sqrt{15}\right)^{2}}$$

The square of \sqrt{15} is 15.

$$\frac{\left(-24\sqrt{3}+12\sqrt{5}\right)\sqrt{15}}{-15}$$

Use the distributive property to multiply -24\sqrt{3}+12\sqrt{5} by \sqrt{15}.

$$\frac{-24\sqrt{3}\sqrt{15}+12\sqrt{5}\sqrt{15}}{-15}$$

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

$$\frac{-24\sqrt{3}\sqrt{3}\sqrt{5}+12\sqrt{5}\sqrt{15}}{-15}$$

Multiply \sqrt{3} and \sqrt{3} to get 3.

$$\frac{-24\times 3\sqrt{5}+12\sqrt{5}\sqrt{15}}{-15}$$

Multiply -24 and 3 to get -72.

$$\frac{-72\sqrt{5}+12\sqrt{5}\sqrt{15}}{-15}$$

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}.

$$\frac{-72\sqrt{5}+12\sqrt{5}\sqrt{5}\sqrt{3}}{-15}$$

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

$$\frac{-72\sqrt{5}+12\times 5\sqrt{3}}{-15}$$

Multiply 12 and 5 to get 60.

$$\frac{-72\sqrt{5}+60\sqrt{3}}{-15}$$