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

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

Evaluate

$$198-72\sqrt{6}\approx 21.63673852$$

Solution Steps

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

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

The square of \sqrt{2} is 2.

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

Multiply 9 and 2 to get 18.

$$\left(18+6\sqrt{2}\sqrt{3}+\left(\sqrt{3}\right)^{2}\right)\left(2\sqrt{3}-3\sqrt{2}\right)^{2}$$

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

$$\left(18+6\sqrt{6}+\left(\sqrt{3}\right)^{2}\right)\left(2\sqrt{3}-3\sqrt{2}\right)^{2}$$

The square of \sqrt{3} is 3.

$$\left(18+6\sqrt{6}+3\right)\left(2\sqrt{3}-3\sqrt{2}\right)^{2}$$

Add 18 and 3 to get 21.

$$\left(21+6\sqrt{6}\right)\left(2\sqrt{3}-3\sqrt{2}\right)^{2}$$

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

$$\left(21+6\sqrt{6}\right)\left(4\left(\sqrt{3}\right)^{2}-12\sqrt{3}\sqrt{2}+9\left(\sqrt{2}\right)^{2}\right)$$

The square of \sqrt{3} is 3.

$$\left(21+6\sqrt{6}\right)\left(4\times 3-12\sqrt{3}\sqrt{2}+9\left(\sqrt{2}\right)^{2}\right)$$

Multiply 4 and 3 to get 12.

$$\left(21+6\sqrt{6}\right)\left(12-12\sqrt{3}\sqrt{2}+9\left(\sqrt{2}\right)^{2}\right)$$

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

$$\left(21+6\sqrt{6}\right)\left(12-12\sqrt{6}+9\left(\sqrt{2}\right)^{2}\right)$$

The square of \sqrt{2} is 2.

$$\left(21+6\sqrt{6}\right)\left(12-12\sqrt{6}+9\times 2\right)$$

Multiply 9 and 2 to get 18.

$$\left(21+6\sqrt{6}\right)\left(12-12\sqrt{6}+18\right)$$

Add 12 and 18 to get 30.

$$\left(21+6\sqrt{6}\right)\left(30-12\sqrt{6}\right)$$

Use the distributive property to multiply 21+6\sqrt{6} by 30-12\sqrt{6} and combine like terms.

$$630-72\sqrt{6}-72\left(\sqrt{6}\right)^{2}$$

The square of \sqrt{6} is 6.

$$630-72\sqrt{6}-72\times 6$$

Multiply -72 and 6 to get -432.

$$630-72\sqrt{6}-432$$

Subtract 432 from 630 to get 198.

$$198-72\sqrt{6}$$

Expand

$$198-72\sqrt{6}$$

Solution Steps

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

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

The square of \sqrt{2} is 2.

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

Multiply 9 and 2 to get 18.

$$\left(18+6\sqrt{2}\sqrt{3}+\left(\sqrt{3}\right)^{2}\right)\left(2\sqrt{3}-3\sqrt{2}\right)^{2}$$

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

$$\left(18+6\sqrt{6}+\left(\sqrt{3}\right)^{2}\right)\left(2\sqrt{3}-3\sqrt{2}\right)^{2}$$

The square of \sqrt{3} is 3.

$$\left(18+6\sqrt{6}+3\right)\left(2\sqrt{3}-3\sqrt{2}\right)^{2}$$

Add 18 and 3 to get 21.

$$\left(21+6\sqrt{6}\right)\left(2\sqrt{3}-3\sqrt{2}\right)^{2}$$

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

$$\left(21+6\sqrt{6}\right)\left(4\left(\sqrt{3}\right)^{2}-12\sqrt{3}\sqrt{2}+9\left(\sqrt{2}\right)^{2}\right)$$

The square of \sqrt{3} is 3.

$$\left(21+6\sqrt{6}\right)\left(4\times 3-12\sqrt{3}\sqrt{2}+9\left(\sqrt{2}\right)^{2}\right)$$

Multiply 4 and 3 to get 12.

$$\left(21+6\sqrt{6}\right)\left(12-12\sqrt{3}\sqrt{2}+9\left(\sqrt{2}\right)^{2}\right)$$

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

$$\left(21+6\sqrt{6}\right)\left(12-12\sqrt{6}+9\left(\sqrt{2}\right)^{2}\right)$$

The square of \sqrt{2} is 2.

$$\left(21+6\sqrt{6}\right)\left(12-12\sqrt{6}+9\times 2\right)$$

Multiply 9 and 2 to get 18.

$$\left(21+6\sqrt{6}\right)\left(12-12\sqrt{6}+18\right)$$

Add 12 and 18 to get 30.

$$\left(21+6\sqrt{6}\right)\left(30-12\sqrt{6}\right)$$

Use the distributive property to multiply 21+6\sqrt{6} by 30-12\sqrt{6} and combine like terms.

$$630-72\sqrt{6}-72\left(\sqrt{6}\right)^{2}$$

The square of \sqrt{6} is 6.

$$630-72\sqrt{6}-72\times 6$$

Multiply -72 and 6 to get -432.

$$630-72\sqrt{6}-432$$

Subtract 432 from 630 to get 198.

$$198-72\sqrt{6}$$