Solve sqrt{15}div(1/sqrt{3}+1/sqrt{5})

$$\sqrt{ 15 } \div ( \frac{ 1 }{ \sqrt{ 3 } } + \frac{ 1 }{ \sqrt{ 5 } } )$$

Evaluate

$$\frac{15\left(\sqrt{5}-\sqrt{3}\right)}{2}\approx 3.780128774$$

Short Solution Steps

Rationalize the denominator of \frac{1}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

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

The square of \sqrt{3} is 3.

$$\frac{\sqrt{15}}{\frac{\sqrt{3}}{3}+\frac{1}{\sqrt{5}}}$$

Rationalize the denominator of \frac{1}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

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

The square of \sqrt{5} is 5.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3 and 5 is 15. Multiply \frac{\sqrt{3}}{3} times \frac{5}{5}. Multiply \frac{\sqrt{5}}{5} times \frac{3}{3}.

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

Since \frac{5\sqrt{3}}{15} and \frac{3\sqrt{5}}{15} have the same denominator, add them by adding their numerators.

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

Divide \sqrt{15} by \frac{5\sqrt{3}+3\sqrt{5}}{15} by multiplying \sqrt{15} by the reciprocal of \frac{5\sqrt{3}+3\sqrt{5}}{15}.

$$\frac{\sqrt{15}\times 15}{5\sqrt{3}+3\sqrt{5}}$$

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

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

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

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

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

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

Calculate 5 to the power of 2 and get 25.

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

The square of \sqrt{3} is 3.

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

Multiply 25 and 3 to get 75.

$$\frac{\sqrt{15}\times 15\left(5\sqrt{3}-3\sqrt{5}\right)}{75-\left(3\sqrt{5}\right)^{2}}$$

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

$$\frac{\sqrt{15}\times 15\left(5\sqrt{3}-3\sqrt{5}\right)}{75-3^{2}\left(\sqrt{5}\right)^{2}}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{\sqrt{15}\times 15\left(5\sqrt{3}-3\sqrt{5}\right)}{75-9\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{\sqrt{15}\times 15\left(5\sqrt{3}-3\sqrt{5}\right)}{75-9\times 5}$$

Multiply 9 and 5 to get 45.

$$\frac{\sqrt{15}\times 15\left(5\sqrt{3}-3\sqrt{5}\right)}{75-45}$$

Subtract 45 from 75 to get 30.

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

Divide \sqrt{15}\times 15\left(5\sqrt{3}-3\sqrt{5}\right) by 30 to get \sqrt{15}\times \frac{1}{2}\left(5\sqrt{3}-3\sqrt{5}\right).

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

Use the distributive property to multiply \sqrt{15}\times \frac{1}{2} by 5\sqrt{3}-3\sqrt{5}.

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

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}.

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

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

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

Multiply 3 and \frac{1}{2} to get \frac{3}{2}.

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

Express \frac{3}{2}\times 5 as a single fraction.

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

Multiply 3 and 5 to get 15.

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

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}.

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

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

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

Multiply 5 and \frac{1}{2} to get \frac{5}{2}.

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

Express \frac{5}{2}\left(-3\right) as a single fraction.

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

Multiply 5 and -3 to get -15.

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

Fraction \frac{-15}{2} can be rewritten as -\frac{15}{2} by extracting the negative sign.

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