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

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

Evaluate

$$-\frac{\sqrt{10}\left(\sqrt{15}+1\right)}{14}\approx -1.100694741$$

Solution Steps

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

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

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

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

The square of \sqrt{5} is 5.

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

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

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

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

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

The square of \sqrt{3} is 3.

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

Multiply 25 and 3 to get 75.

$$\frac{5\sqrt{2}\left(\sqrt{5}+5\sqrt{3}\right)}{5-75}$$

Subtract 75 from 5 to get -70.

$$\frac{5\sqrt{2}\left(\sqrt{5}+5\sqrt{3}\right)}{-70}$$

Divide 5\sqrt{2}\left(\sqrt{5}+5\sqrt{3}\right) by -70 to get -\frac{1}{14}\sqrt{2}\left(\sqrt{5}+5\sqrt{3}\right).

$$-\frac{1}{14}\sqrt{2}\left(\sqrt{5}+5\sqrt{3}\right)$$

Use the distributive property to multiply -\frac{1}{14}\sqrt{2} by \sqrt{5}+5\sqrt{3}.

$$-\frac{1}{14}\sqrt{2}\sqrt{5}-\frac{1}{14}\sqrt{2}\times 5\sqrt{3}$$

To multiply \sqrt{2} and \sqrt{5}, multiply the numbers under the square root.

$$-\frac{1}{14}\sqrt{10}-\frac{1}{14}\sqrt{2}\times 5\sqrt{3}$$

Express -\frac{1}{14}\times 5 as a single fraction.

$$-\frac{1}{14}\sqrt{10}+\frac{-5}{14}\sqrt{2}\sqrt{3}$$

Fraction \frac{-5}{14} can be rewritten as -\frac{5}{14} by extracting the negative sign.

$$-\frac{1}{14}\sqrt{10}-\frac{5}{14}\sqrt{2}\sqrt{3}$$

To multiply \sqrt{2} and \sqrt{3}, multiply the numbers under the square root.

$$-\frac{1}{14}\sqrt{10}-\frac{5}{14}\sqrt{6}$$