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

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

Evaluate

$$92-16\sqrt{15}\approx 30.032266461$$

Solution Steps

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

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

The square of \sqrt{5} is 5.

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

Multiply 16 and 5 to get 80.

$$80-16\sqrt{5}\sqrt{3}+4\left(\sqrt{3}\right)^{2}$$

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

$$80-16\sqrt{15}+4\left(\sqrt{3}\right)^{2}$$

The square of \sqrt{3} is 3.

$$80-16\sqrt{15}+4\times 3$$

Multiply 4 and 3 to get 12.

$$80-16\sqrt{15}+12$$

Add 80 and 12 to get 92.

$$92-16\sqrt{15}$$

Expand

$$92-16\sqrt{15}$$

Solution Steps

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

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

The square of \sqrt{5} is 5.

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

Multiply 16 and 5 to get 80.

$$80-16\sqrt{5}\sqrt{3}+4\left(\sqrt{3}\right)^{2}$$

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

$$80-16\sqrt{15}+4\left(\sqrt{3}\right)^{2}$$

The square of \sqrt{3} is 3.

$$80-16\sqrt{15}+4\times 3$$

Multiply 4 and 3 to get 12.

$$80-16\sqrt{15}+12$$

Add 80 and 12 to get 92.

$$92-16\sqrt{15}$$