Solve 15/2sqrt{5+5}

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

Evaluate

$$15-6\sqrt{5}\approx 1.583592135$$

Solution Steps

Rationalize the denominator of \frac{15}{2\sqrt{5}+5} by multiplying numerator and denominator by 2\sqrt{5}-5.

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

Consider \left(2\sqrt{5}+5\right)\left(2\sqrt{5}-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}.

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

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

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

Calculate 2 to the power of 2 and get 4.

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

The square of \sqrt{5} is 5.

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

Multiply 4 and 5 to get 20.

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

Calculate 5 to the power of 2 and get 25.

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

Subtract 25 from 20 to get -5.

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

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

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

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

$$-6\sqrt{5}+15$$