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Chapter 2: Problem 35

Perform the indicated operations and write the result in standard form. $$(-3-\sqrt{-7})^{2}$$

Short Answer

Expert verified
The result of the operation \((-3-\sqrt{-7})^{2}\) is \(2 + 6i\sqrt{7}\).

Step by step solution

01

Recognize the complex number

Recognize that \(\sqrt{-7}\) means that the complex number involved is \(i\sqrt{7}\), where i is the imaginary unit.
02

Write out the complex number in standard form

Write out the whole expression replacing \(\sqrt{-7}\) with \(i\sqrt{7}\), so the expression is now \((-3-i\sqrt{7})^{2}\).
03

Expand the square

Expand the square using the FOIL method or the identity \((a-b)^{2}=a^{2}-2ab+b^{2}\). So, \((-3-i\sqrt{7})^{2} = (-3)^2 - 2*(-3)*i*\sqrt{7} + (i*\sqrt{7})^2\).
04

Simplify Each Part

The previous expression simplifies to \(9 + 6i\sqrt{7} - 7\).
05

Further Simplify and Write in Standard Form

Simplify the real numbers, leading to \(2 + 6i\sqrt{7}\), which represents the complex number in a+bi form, where a and b are the real numbers.

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