Solve frac{2sqrt{10}}{-8-4sqrt{3}}

$$\frac { 2 \sqrt { 10 } } { - 8 - 4 \sqrt { 3 } }$$

Evaluate

$$\frac{\sqrt{30}}{2}-\sqrt{10}\approx -0.423664873$$

Solution Steps

Rationalize the denominator of \frac{2\sqrt{10}}{-8-4\sqrt{3}} by multiplying numerator and denominator by -8+4\sqrt{3}.

$$\frac{2\sqrt{10}\left(-8+4\sqrt{3}\right)}{\left(-8-4\sqrt{3}\right)\left(-8+4\sqrt{3}\right)}$$

Consider \left(-8-4\sqrt{3}\right)\left(-8+4\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}.

$$\frac{2\sqrt{10}\left(-8+4\sqrt{3}\right)}{\left(-8\right)^{2}-\left(-4\sqrt{3}\right)^{2}}$$

Calculate -8 to the power of 2 and get 64.

$$\frac{2\sqrt{10}\left(-8+4\sqrt{3}\right)}{64-\left(-4\sqrt{3}\right)^{2}}$$

Expand \left(-4\sqrt{3}\right)^{2}.

$$\frac{2\sqrt{10}\left(-8+4\sqrt{3}\right)}{64-\left(-4\right)^{2}\left(\sqrt{3}\right)^{2}}$$

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

$$\frac{2\sqrt{10}\left(-8+4\sqrt{3}\right)}{64-16\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{2\sqrt{10}\left(-8+4\sqrt{3}\right)}{64-16\times 3}$$

Multiply 16 and 3 to get 48.

$$\frac{2\sqrt{10}\left(-8+4\sqrt{3}\right)}{64-48}$$

Subtract 48 from 64 to get 16.

$$\frac{2\sqrt{10}\left(-8+4\sqrt{3}\right)}{16}$$

Divide 2\sqrt{10}\left(-8+4\sqrt{3}\right) by 16 to get \frac{1}{8}\sqrt{10}\left(-8+4\sqrt{3}\right).

$$\frac{1}{8}\sqrt{10}\left(-8+4\sqrt{3}\right)$$

Use the distributive property to multiply \frac{1}{8}\sqrt{10} by -8+4\sqrt{3}.

$$\frac{1}{8}\sqrt{10}\left(-8\right)+\frac{1}{8}\sqrt{10}\times 4\sqrt{3}$$

Multiply \frac{1}{8} and -8 to get \frac{-8}{8}.

$$\frac{-8}{8}\sqrt{10}+\frac{1}{8}\sqrt{10}\times 4\sqrt{3}$$

Divide -8 by 8 to get -1.

$$-\sqrt{10}+\frac{1}{8}\sqrt{10}\times 4\sqrt{3}$$

Multiply \frac{1}{8} and 4 to get \frac{4}{8}.

$$-\sqrt{10}+\frac{4}{8}\sqrt{10}\sqrt{3}$$

Reduce the fraction \frac{4}{8} to lowest terms by extracting and canceling out 4.

$$-\sqrt{10}+\frac{1}{2}\sqrt{10}\sqrt{3}$$

To multiply \sqrt{10} and \sqrt{3}, multiply the numbers under the square root.

$$-\sqrt{10}+\frac{1}{2}\sqrt{30}$$