Solve (-2-sqrt{-7})(2+sqrt{-7)}

$$(-2- \sqrt{ -7 } )(2+ \sqrt{ -7) }$$

Evaluate (complex solution)

$$-4\sqrt{7}i+3\approx 3-10.583005244i$$

Solution Steps

Factor -7=7\left(-1\right). Rewrite the square root of the product \sqrt{7\left(-1\right)} as the product of square roots \sqrt{7}\sqrt{-1}. By definition, the square root of -1 is i.

$$\left(-2-\sqrt{7}i\right)\left(2+\sqrt{-7}\right)$$

Multiply -1 and i to get -i.

$$\left(-2-i\sqrt{7}\right)\left(2+\sqrt{-7}\right)$$

Factor -7=7\left(-1\right). Rewrite the square root of the product \sqrt{7\left(-1\right)} as the product of square roots \sqrt{7}\sqrt{-1}. By definition, the square root of -1 is i.

$$\left(-2-i\sqrt{7}\right)\left(2+\sqrt{7}i\right)$$

Apply the distributive property by multiplying each term of -2-i\sqrt{7} by each term of 2+\sqrt{7}i.

$$-4-2\sqrt{7}i-2i\sqrt{7}+\left(\sqrt{7}\right)^{2}$$

Multiply -2 and i to get -2i.

$$-4-2i\sqrt{7}-2i\sqrt{7}+\left(\sqrt{7}\right)^{2}$$

Combine -2i\sqrt{7} and -2i\sqrt{7} to get -4i\sqrt{7}.

$$-4-4i\sqrt{7}+\left(\sqrt{7}\right)^{2}$$

The square of \sqrt{7} is 7.

$$-4-4i\sqrt{7}+7$$

Add -4 and 7 to get 3.

$$3-4i\sqrt{7}$$

Evaluate

$$\text{Indeterminate}$$

Solution Steps

Apply the distributive property by multiplying each term of -2-\sqrt{-7} by each term of 2+\sqrt{-7}.

$$-4-2\sqrt{-7}-2\sqrt{-7}-\left(\sqrt{-7}\right)^{2}$$

Combine -2\sqrt{-7} and -2\sqrt{-7} to get -4\sqrt{-7}.

$$-4-4\sqrt{-7}-\left(\sqrt{-7}\right)^{2}$$

Calculate \sqrt{-7} to the power of 2 and get -7.

$$-4-4\sqrt{-7}-\left(-7\right)$$

Multiply -1 and -7 to get 7.

$$-4-4\sqrt{-7}+7$$

Add -4 and 7 to get 3.

$$3-4\sqrt{-7}$$

Real Part (complex solution)

$$3$$