Solve -9/2sqrt{5-3}

$$\frac { - 9 } { 2 \sqrt { 5 } - 3 }$$

Evaluate

$$\frac{-18\sqrt{5}-27}{11}\approx -6.113565781$$

Solution Steps

Rationalize the denominator of \frac{-9}{2\sqrt{5}-3} by multiplying numerator and denominator by 2\sqrt{5}+3.

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

Consider \left(2\sqrt{5}-3\right)\left(2\sqrt{5}+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}.

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

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

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

Calculate 2 to the power of 2 and get 4.

$$\frac{-9\left(2\sqrt{5}+3\right)}{4\left(\sqrt{5}\right)^{2}-3^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{-9\left(2\sqrt{5}+3\right)}{4\times 5-3^{2}}$$

Multiply 4 and 5 to get 20.

$$\frac{-9\left(2\sqrt{5}+3\right)}{20-3^{2}}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{-9\left(2\sqrt{5}+3\right)}{20-9}$$

Subtract 9 from 20 to get 11.

$$\frac{-9\left(2\sqrt{5}+3\right)}{11}$$

Use the distributive property to multiply -9 by 2\sqrt{5}+3.

$$\frac{-18\sqrt{5}-27}{11}$$