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

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

Evaluate

$$7$$

Solution Steps

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

$$16\left(\sqrt{3}\right)^{2}-24\sqrt{3}\sqrt{2}+9\left(\sqrt{2}\right)^{2}-\left(3\sqrt{3}-4\sqrt{2}\right)^{2}$$

The square of \sqrt{3} is 3.

$$16\times 3-24\sqrt{3}\sqrt{2}+9\left(\sqrt{2}\right)^{2}-\left(3\sqrt{3}-4\sqrt{2}\right)^{2}$$

Multiply 16 and 3 to get 48.

$$48-24\sqrt{3}\sqrt{2}+9\left(\sqrt{2}\right)^{2}-\left(3\sqrt{3}-4\sqrt{2}\right)^{2}$$

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

$$48-24\sqrt{6}+9\left(\sqrt{2}\right)^{2}-\left(3\sqrt{3}-4\sqrt{2}\right)^{2}$$

The square of \sqrt{2} is 2.

$$48-24\sqrt{6}+9\times 2-\left(3\sqrt{3}-4\sqrt{2}\right)^{2}$$

Multiply 9 and 2 to get 18.

$$48-24\sqrt{6}+18-\left(3\sqrt{3}-4\sqrt{2}\right)^{2}$$

Add 48 and 18 to get 66.

$$66-24\sqrt{6}-\left(3\sqrt{3}-4\sqrt{2}\right)^{2}$$

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

$$66-24\sqrt{6}-\left(9\left(\sqrt{3}\right)^{2}-24\sqrt{3}\sqrt{2}+16\left(\sqrt{2}\right)^{2}\right)$$

The square of \sqrt{3} is 3.

$$66-24\sqrt{6}-\left(9\times 3-24\sqrt{3}\sqrt{2}+16\left(\sqrt{2}\right)^{2}\right)$$

Multiply 9 and 3 to get 27.

$$66-24\sqrt{6}-\left(27-24\sqrt{3}\sqrt{2}+16\left(\sqrt{2}\right)^{2}\right)$$

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

$$66-24\sqrt{6}-\left(27-24\sqrt{6}+16\left(\sqrt{2}\right)^{2}\right)$$

The square of \sqrt{2} is 2.

$$66-24\sqrt{6}-\left(27-24\sqrt{6}+16\times 2\right)$$

Multiply 16 and 2 to get 32.

$$66-24\sqrt{6}-\left(27-24\sqrt{6}+32\right)$$

Add 27 and 32 to get 59.

$$66-24\sqrt{6}-\left(59-24\sqrt{6}\right)$$

To find the opposite of 59-24\sqrt{6}, find the opposite of each term.

$$66-24\sqrt{6}-59+24\sqrt{6}$$

Subtract 59 from 66 to get 7.

$$7-24\sqrt{6}+24\sqrt{6}$$

Combine -24\sqrt{6} and 24\sqrt{6} to get 0.

$$7$$

Factor

$$7$$