Solve {left(2sqrt{3}+3sqrt{5}right)}^2

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

Evaluate

$$12\sqrt{15}+57\approx 103.475800154$$

Solution Steps

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

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

The square of \sqrt{3} is 3.

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

Multiply 4 and 3 to get 12.

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

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

$$12+12\sqrt{15}+9\left(\sqrt{5}\right)^{2}$$

The square of \sqrt{5} is 5.

$$12+12\sqrt{15}+9\times 5$$

Multiply 9 and 5 to get 45.

$$12+12\sqrt{15}+45$$

Add 12 and 45 to get 57.

$$57+12\sqrt{15}$$

Expand

$$12 \sqrt{15} + 57 = 103.475800154$$

Solution Steps

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

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

The square of \sqrt{3} is 3.

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

Multiply 4 and 3 to get 12.

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

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

$$12+12\sqrt{15}+9\left(\sqrt{5}\right)^{2}$$

The square of \sqrt{5} is 5.

$$12+12\sqrt{15}+9\times 5$$

Multiply 9 and 5 to get 45.

$$12+12\sqrt{15}+45$$

Add 12 and 45 to get 57.

$$57+12\sqrt{15}$$