Solve (1/2sqrt{3}-4sqrt{5})^2=

$$( \frac { 1 } { 2 } \sqrt { 3 } - 4 \sqrt { 5 } ) ^ { 2 } =$$

Evaluate

$$\frac{323}{4}-4\sqrt{15}\approx 65.258066615$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\frac{1}{2}\sqrt{3}-4\sqrt{5}\right)^{2}.

$$\frac{1}{4}\left(\sqrt{3}\right)^{2}-4\sqrt{3}\sqrt{5}+16\left(\sqrt{5}\right)^{2}$$

The square of \sqrt{3} is 3.

$$\frac{1}{4}\times 3-4\sqrt{3}\sqrt{5}+16\left(\sqrt{5}\right)^{2}$$

Multiply \frac{1}{4} and 3 to get \frac{3}{4}.

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

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

$$\frac{3}{4}-4\sqrt{15}+16\left(\sqrt{5}\right)^{2}$$

The square of \sqrt{5} is 5.

$$\frac{3}{4}-4\sqrt{15}+16\times 5$$

Multiply 16 and 5 to get 80.

$$\frac{3}{4}-4\sqrt{15}+80$$

Add \frac{3}{4} and 80 to get \frac{323}{4}.

$$\frac{323}{4}-4\sqrt{15}$$

Expand

$$\frac{323}{4} - 4 \sqrt{15} = 65.258066615$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\frac{1}{2}\sqrt{3}-4\sqrt{5}\right)^{2}.

$$\frac{1}{4}\left(\sqrt{3}\right)^{2}-4\sqrt{3}\sqrt{5}+16\left(\sqrt{5}\right)^{2}$$

The square of \sqrt{3} is 3.

$$\frac{1}{4}\times 3-4\sqrt{3}\sqrt{5}+16\left(\sqrt{5}\right)^{2}$$

Multiply \frac{1}{4} and 3 to get \frac{3}{4}.

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

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

$$\frac{3}{4}-4\sqrt{15}+16\left(\sqrt{5}\right)^{2}$$

The square of \sqrt{5} is 5.

$$\frac{3}{4}-4\sqrt{15}+16\times 5$$

Multiply 16 and 5 to get 80.

$$\frac{3}{4}-4\sqrt{15}+80$$

Add \frac{3}{4} and 80 to get \frac{323}{4}.

$$\frac{323}{4}-4\sqrt{15}$$