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Chapter 2: Problem 34

Perform the indicated operations and write the result in standard form. $$(-5-\sqrt{-9})^{2}$$

Short Answer

Expert verified
The result in standard form is \( 16 + 30i \)

Step by step solution

01

Recognizing the Complex Number

The first step is to recognize the complex number in the exercise. A complex number comes in the form \( a + bi \) where \( a \) and \( b \) are real numbers and \( i \) is the imaginary unit with the property that \( i^{2} = -1 \). It means that \( \sqrt{-9} = 3i \) since \( i^{2} = -1 \). So, the complex number given here is \( -5 - 3i \).
02

Squaring the Complex Number

Squaring the complex number means multiplying it by itself. By using the formula for squaring a binomial \( (a-b)^{2} = a^{2} - 2ab + b^{2} \), we can square \( -5 - 3i \). Hence, \( (-5-3i)^{2} = (-5)^{2} - 2* -5 * 3i + (3i)^{2} \).
03

Computing the Result

Now, we just compute the result. The square of \( -5 \) equals 25. The product \( -2 * -5 * 3i \) equals \( 30i \). The square of \( 3i \) equals \( (3i)^{2} = 9i^{2} = 9*-1 = -9 \). Therefore, the result is \( 25 + 30i -9 = 16 + 30i \).

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