Solve 5/3sqrt{5-2}

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

Evaluate

$$\frac{15\sqrt{5}+10}{41}\approx 1.061976089$$

Solution Steps

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

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

Consider \left(3\sqrt{5}-2\right)\left(3\sqrt{5}+2\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

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

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

Calculate 3 to the power of 2 and get 9.

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

The square of \sqrt{5} is 5.

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

Multiply 9 and 5 to get 45.

$$\frac{5\left(3\sqrt{5}+2\right)}{45-2^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{5\left(3\sqrt{5}+2\right)}{45-4}$$

Subtract 4 from 45 to get 41.

$$\frac{5\left(3\sqrt{5}+2\right)}{41}$$

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

$$\frac{15\sqrt{5}+10}{41}$$