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)

$$4-2\sqrt{6}\approx -0.898979486$$

Solution Steps

Calculate the square root of -1 and get i.

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

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.

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

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.

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

Multiply -1 and i to get -i.

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

Calculate the square root of -1 and get i.

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

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.

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

Multiply -1 and i to get -i.

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

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.

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

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

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

Multiply i and i to get -1.

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

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.

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

Add -1 and 2 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.

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

Add 1 and 3 to get 4.

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

Real Part (complex solution)

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

Evaluate

$$\text{Indeterminate}$$