Solve frac{3-2sqrt{5}}{2+5sqrt{3}}

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

Evaluate

$$\frac{4\sqrt{5}+15\sqrt{3}-10\sqrt{15}-6}{71}\approx -0.138095767$$

Solution Steps

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

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

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

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

Calculate 2 to the power of 2 and get 4.

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

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

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

Calculate 5 to the power of 2 and get 25.

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

The square of \sqrt{3} is 3.

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

Multiply 25 and 3 to get 75.

$$\frac{\left(3-2\sqrt{5}\right)\left(2-5\sqrt{3}\right)}{4-75}$$

Subtract 75 from 4 to get -71.

$$\frac{\left(3-2\sqrt{5}\right)\left(2-5\sqrt{3}\right)}{-71}$$

Apply the distributive property by multiplying each term of 3-2\sqrt{5} by each term of 2-5\sqrt{3}.

$$\frac{6-15\sqrt{3}-4\sqrt{5}+10\sqrt{3}\sqrt{5}}{-71}$$

To multiply \sqrt{3} and \sqrt{5}, multiply the numbers under the square root.

$$\frac{6-15\sqrt{3}-4\sqrt{5}+10\sqrt{15}}{-71}$$

Multiply both numerator and denominator by -1.

$$\frac{-6+15\sqrt{3}+4\sqrt{5}-10\sqrt{15}}{71}$$