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

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

Evaluate

$$47-12\sqrt{15}\approx 0.524199846$$

Solution Steps

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

$$4\left(\sqrt{5}\right)^{2}-12\sqrt{5}\sqrt{3}+9\left(\sqrt{3}\right)^{2}$$

The square of \sqrt{5} is 5.

$$4\times 5-12\sqrt{5}\sqrt{3}+9\left(\sqrt{3}\right)^{2}$$

Multiply 4 and 5 to get 20.

$$20-12\sqrt{5}\sqrt{3}+9\left(\sqrt{3}\right)^{2}$$

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

$$20-12\sqrt{15}+9\left(\sqrt{3}\right)^{2}$$

The square of \sqrt{3} is 3.

$$20-12\sqrt{15}+9\times 3$$

Multiply 9 and 3 to get 27.

$$20-12\sqrt{15}+27$$

Add 20 and 27 to get 47.

$$47-12\sqrt{15}$$

Expand

$$47-12\sqrt{15}$$

Solution Steps

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

$$4\left(\sqrt{5}\right)^{2}-12\sqrt{5}\sqrt{3}+9\left(\sqrt{3}\right)^{2}$$

The square of \sqrt{5} is 5.

$$4\times 5-12\sqrt{5}\sqrt{3}+9\left(\sqrt{3}\right)^{2}$$

Multiply 4 and 5 to get 20.

$$20-12\sqrt{5}\sqrt{3}+9\left(\sqrt{3}\right)^{2}$$

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

$$20-12\sqrt{15}+9\left(\sqrt{3}\right)^{2}$$

The square of \sqrt{3} is 3.

$$20-12\sqrt{15}+9\times 3$$

Multiply 9 and 3 to get 27.

$$20-12\sqrt{15}+27$$

Add 20 and 27 to get 47.

$$47-12\sqrt{15}$$