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

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

Evaluate

$$12\sqrt{6}+29\approx 58.393876913$$

Solution Steps

Consider \left(2\sqrt{3}+3\sqrt{2}\right)\left(2\sqrt{3}-3\sqrt{2}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Expand \left(2\sqrt{3}\right)^{2}.

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

Calculate 2 to the power of 2 and get 4.

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

The square of \sqrt{3} is 3.

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

Multiply 4 and 3 to get 12.

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

Expand \left(3\sqrt{2}\right)^{2}.

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

Calculate 3 to the power of 2 and get 9.

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

The square of \sqrt{2} is 2.

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

Multiply 9 and 2 to get 18.

$$12-18+\left(2\sqrt{2}+3\sqrt{3}\right)^{2}$$

Subtract 18 from 12 to get -6.

$$-6+\left(2\sqrt{2}+3\sqrt{3}\right)^{2}$$

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

$$-6+4\left(\sqrt{2}\right)^{2}+12\sqrt{2}\sqrt{3}+9\left(\sqrt{3}\right)^{2}$$

The square of \sqrt{2} is 2.

$$-6+4\times 2+12\sqrt{2}\sqrt{3}+9\left(\sqrt{3}\right)^{2}$$

Multiply 4 and 2 to get 8.

$$-6+8+12\sqrt{2}\sqrt{3}+9\left(\sqrt{3}\right)^{2}$$

To multiply \sqrt{2} and \sqrt{3}, multiply the numbers under the square root.

$$-6+8+12\sqrt{6}+9\left(\sqrt{3}\right)^{2}$$

The square of \sqrt{3} is 3.

$$-6+8+12\sqrt{6}+9\times 3$$

Multiply 9 and 3 to get 27.

$$-6+8+12\sqrt{6}+27$$

Add 8 and 27 to get 35.

$$-6+35+12\sqrt{6}$$

Add -6 and 35 to get 29.

$$29+12\sqrt{6}$$