Solve 3/5-4sqrt{5}

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

Evaluate

$$-\frac{12\sqrt{5}}{55}-\frac{3}{11}\approx -0.76059665$$

Solution Steps

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

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

Consider \left(5-4\sqrt{5}\right)\left(5+4\sqrt{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}.

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

Calculate 5 to the power of 2 and get 25.

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

Expand \left(-4\sqrt{5}\right)^{2}.

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

Calculate -4 to the power of 2 and get 16.

$$\frac{3\left(5+4\sqrt{5}\right)}{25-16\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{3\left(5+4\sqrt{5}\right)}{25-16\times 5}$$

Multiply 16 and 5 to get 80.

$$\frac{3\left(5+4\sqrt{5}\right)}{25-80}$$

Subtract 80 from 25 to get -55.

$$\frac{3\left(5+4\sqrt{5}\right)}{-55}$$

Use the distributive property to multiply 3 by 5+4\sqrt{5}.

$$\frac{15+12\sqrt{5}}{-55}$$