Solve 4/5sqrt{5-5sqrt{3}}

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

Evaluate

$$\frac{2\left(\sqrt{3}+\sqrt{5}\right)}{5}\approx 1.587247514$$

Solution Steps

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

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

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

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

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

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

Calculate 5 to the power of 2 and get 25.

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

The square of \sqrt{5} is 5.

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

Multiply 25 and 5 to get 125.

$$\frac{4\left(5\sqrt{5}+5\sqrt{3}\right)}{125-\left(-5\sqrt{3}\right)^{2}}$$

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

$$\frac{4\left(5\sqrt{5}+5\sqrt{3}\right)}{125-\left(-5\right)^{2}\left(\sqrt{3}\right)^{2}}$$

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

$$\frac{4\left(5\sqrt{5}+5\sqrt{3}\right)}{125-25\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{4\left(5\sqrt{5}+5\sqrt{3}\right)}{125-25\times 3}$$

Multiply 25 and 3 to get 75.

$$\frac{4\left(5\sqrt{5}+5\sqrt{3}\right)}{125-75}$$

Subtract 75 from 125 to get 50.

$$\frac{4\left(5\sqrt{5}+5\sqrt{3}\right)}{50}$$

Divide 4\left(5\sqrt{5}+5\sqrt{3}\right) by 50 to get \frac{2}{25}\left(5\sqrt{5}+5\sqrt{3}\right).

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

Use the distributive property to multiply \frac{2}{25} by 5\sqrt{5}+5\sqrt{3}.

$$\frac{2}{25}\times 5\sqrt{5}+\frac{2}{25}\times 5\sqrt{3}$$

Express \frac{2}{25}\times 5 as a single fraction.

$$\frac{2\times 5}{25}\sqrt{5}+\frac{2}{25}\times 5\sqrt{3}$$

Multiply 2 and 5 to get 10.

$$\frac{10}{25}\sqrt{5}+\frac{2}{25}\times 5\sqrt{3}$$

Reduce the fraction \frac{10}{25} to lowest terms by extracting and canceling out 5.

$$\frac{2}{5}\sqrt{5}+\frac{2}{25}\times 5\sqrt{3}$$

Express \frac{2}{25}\times 5 as a single fraction.

$$\frac{2}{5}\sqrt{5}+\frac{2\times 5}{25}\sqrt{3}$$

Multiply 2 and 5 to get 10.

$$\frac{2}{5}\sqrt{5}+\frac{10}{25}\sqrt{3}$$

Reduce the fraction \frac{10}{25} to lowest terms by extracting and canceling out 5.

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