Solve -(sqrt{-1}+sqrt{-2}-sqrt{-3})(sqrt{-1}-sqrt{-2}+sqrt{-3})

$$- ( \sqrt { - 1 } + \sqrt { - 2 } - \sqrt { - 3 } ) ( \sqrt { - 1 } - \sqrt { - 2 } + \sqrt { - 3 } )$$

Evaluate (complex solution)

$$2\left(\sqrt{6}-2\right)\approx 0.898979486$$

Solution Steps

Calculate the square root of -1 and get i.

$$\left(-\left(i+\sqrt{-2}-\sqrt{-3}\right)\right)\left(\sqrt{-1}-\sqrt{-2}+\sqrt{-3}\right)$$

Factor -2=2\left(-1\right). Rewrite the square root of the product \sqrt{2\left(-1\right)} as the product of square roots \sqrt{2}\sqrt{-1}. By definition, the square root of -1 is i.

$$\left(-\left(i+\sqrt{2}i-\sqrt{-3}\right)\right)\left(\sqrt{-1}-\sqrt{-2}+\sqrt{-3}\right)$$

Factor -3=3\left(-1\right). Rewrite the square root of the product \sqrt{3\left(-1\right)} as the product of square roots \sqrt{3}\sqrt{-1}. By definition, the square root of -1 is i.

$$\left(-\left(i+\sqrt{2}i-\sqrt{3}i\right)\right)\left(\sqrt{-1}-\sqrt{-2}+\sqrt{-3}\right)$$

Multiply -1 and i to get -i.

$$\left(-\left(i+\sqrt{2}i-i\sqrt{3}\right)\right)\left(\sqrt{-1}-\sqrt{-2}+\sqrt{-3}\right)$$

To find the opposite of i+\sqrt{2}i-i\sqrt{3}, find the opposite of each term.

$$\left(-i-\sqrt{2}i+i\sqrt{3}\right)\left(\sqrt{-1}-\sqrt{-2}+\sqrt{-3}\right)$$

Multiply -1 and i to get -i.

$$\left(-i-i\sqrt{2}+i\sqrt{3}\right)\left(\sqrt{-1}-\sqrt{-2}+\sqrt{-3}\right)$$

Calculate the square root of -1 and get i.

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

Factor -2=2\left(-1\right). Rewrite the square root of the product \sqrt{2\left(-1\right)} as the product of square roots \sqrt{2}\sqrt{-1}. By definition, the square root of -1 is i.

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

Multiply -1 and i to get -i.

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

Factor -3=3\left(-1\right). Rewrite the square root of the product \sqrt{3\left(-1\right)} as the product of square roots \sqrt{3}\sqrt{-1}. By definition, the square root of -1 is i.

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

Apply the distributive property by multiplying each term of -i-i\sqrt{2}+i\sqrt{3} by each term of i-i\sqrt{2}+\sqrt{3}i.

$$1-\sqrt{2}-i\sqrt{3}i+\sqrt{2}-\left(\sqrt{2}\right)^{2}+\sqrt{3}\sqrt{2}-\sqrt{3}+\sqrt{3}\sqrt{2}-\left(\sqrt{3}\right)^{2}$$

Multiply -i and i to get 1.

$$1-\sqrt{2}+\sqrt{3}+\sqrt{2}-\left(\sqrt{2}\right)^{2}+\sqrt{3}\sqrt{2}-\sqrt{3}+\sqrt{3}\sqrt{2}-\left(\sqrt{3}\right)^{2}$$

Combine -\sqrt{2} and \sqrt{2} to get 0.

$$1+\sqrt{3}-\left(\sqrt{2}\right)^{2}+\sqrt{3}\sqrt{2}-\sqrt{3}+\sqrt{3}\sqrt{2}-\left(\sqrt{3}\right)^{2}$$

The square of \sqrt{2} is 2.

$$1+\sqrt{3}-2+\sqrt{3}\sqrt{2}-\sqrt{3}+\sqrt{3}\sqrt{2}-\left(\sqrt{3}\right)^{2}$$

Subtract 2 from 1 to get -1.

$$-1+\sqrt{3}+\sqrt{3}\sqrt{2}-\sqrt{3}+\sqrt{3}\sqrt{2}-\left(\sqrt{3}\right)^{2}$$

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

$$-1+\sqrt{3}+\sqrt{6}-\sqrt{3}+\sqrt{3}\sqrt{2}-\left(\sqrt{3}\right)^{2}$$

Combine \sqrt{3} and -\sqrt{3} to get 0.

$$-1+\sqrt{6}+\sqrt{3}\sqrt{2}-\left(\sqrt{3}\right)^{2}$$

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

$$-1+\sqrt{6}+\sqrt{6}-\left(\sqrt{3}\right)^{2}$$

Combine \sqrt{6} and \sqrt{6} to get 2\sqrt{6}.

$$-1+2\sqrt{6}-\left(\sqrt{3}\right)^{2}$$

The square of \sqrt{3} is 3.

$$-1+2\sqrt{6}-3$$

Subtract 3 from -1 to get -4.

$$-4+2\sqrt{6}$$

Real Part (complex solution)

$$2 {(\sqrt{6} - 2)} = 0.898979486$$

Evaluate

$$\text{Indeterminate}$$