Solve frac{64+96-(4sqrt{3}+4)^2}{16*4sqrt{5}}=

$$\frac { 64 + 96 - ( 4 \sqrt { 3 } + 4 ) ^ { 2 } } { 16 \times 4 \sqrt { 5 } } =$$

Evaluate

$$\frac{\sqrt{15}\left(\sqrt{3}-1\right)}{10}\approx 0.283522059$$

Solution Steps

Add 64 and 96 to get 160.

$$\frac{160-\left(4\sqrt{3}+4\right)^{2}}{4\times 16\sqrt{5}}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(4\sqrt{3}+4\right)^{2}.

$$\frac{160-\left(16\left(\sqrt{3}\right)^{2}+32\sqrt{3}+16\right)}{4\times 16\sqrt{5}}$$

The square of \sqrt{3} is 3.

$$\frac{160-\left(16\times 3+32\sqrt{3}+16\right)}{4\times 16\sqrt{5}}$$

Multiply 16 and 3 to get 48.

$$\frac{160-\left(48+32\sqrt{3}+16\right)}{4\times 16\sqrt{5}}$$

Add 48 and 16 to get 64.

$$\frac{160-\left(64+32\sqrt{3}\right)}{4\times 16\sqrt{5}}$$

To find the opposite of 64+32\sqrt{3}, find the opposite of each term.

$$\frac{160-64-32\sqrt{3}}{4\times 16\sqrt{5}}$$

Subtract 64 from 160 to get 96.

$$\frac{96-32\sqrt{3}}{4\times 16\sqrt{5}}$$

Multiply 4 and 16 to get 64.

$$\frac{96-32\sqrt{3}}{64\sqrt{5}}$$

Rationalize the denominator of \frac{96-32\sqrt{3}}{64\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{\left(96-32\sqrt{3}\right)\sqrt{5}}{64\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{\left(96-32\sqrt{3}\right)\sqrt{5}}{64\times 5}$$

Multiply 64 and 5 to get 320.

$$\frac{\left(96-32\sqrt{3}\right)\sqrt{5}}{320}$$

Use the distributive property to multiply 96-32\sqrt{3} by \sqrt{5}.

$$\frac{96\sqrt{5}-32\sqrt{3}\sqrt{5}}{320}$$

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

$$\frac{96\sqrt{5}-32\sqrt{15}}{320}$$

Factor

$$\frac{\sqrt{3} \sqrt{5} {(\sqrt{3} - 1)}}{10} = 0.28352205862919516$$