Solve 5/sqrt{5-5}

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

Evaluate

$$\frac{-\sqrt{5}-5}{4}\approx -1.809016994$$

Solution Steps

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

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

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

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

Square \sqrt{5}. Square 5.

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

Subtract 25 from 5 to get -20.

$$\frac{5\left(\sqrt{5}+5\right)}{-20}$$

Divide 5\left(\sqrt{5}+5\right) by -20 to get -\frac{1}{4}\left(\sqrt{5}+5\right).

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

Use the distributive property to multiply -\frac{1}{4} by \sqrt{5}+5.

$$-\frac{1}{4}\sqrt{5}-\frac{1}{4}\times 5$$

Express -\frac{1}{4}\times 5 as a single fraction.

$$-\frac{1}{4}\sqrt{5}+\frac{-5}{4}$$

Fraction \frac{-5}{4} can be rewritten as -\frac{5}{4} by extracting the negative sign.

$$-\frac{1}{4}\sqrt{5}-\frac{5}{4}$$