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Chapter 2: Problem 67

Write the complex number in standard form. $$(\sqrt{-15})^{2}$$

Short Answer

Expert verified
-15

Step by step solution

01

Understanding and Breaking Down the Given Problem

The task is to write the number \((\sqrt{-15})^{2}\) in standard form. The standard form of a complex number is a + bi, where a and b are real numbers, and i is the imaginary unit with the property that \(i = \sqrt{-1}\). In this expression, the presence of a negative number under the square root sign indicates that the result will be a pure imaginary number (there's no real part).
02

Calculation

Start by looking at the square root part. \(\sqrt{-15}\) is equal to \(\sqrt{15} * \sqrt{-1}\). We know that the square root of -1 is denoted by 'i', that is, \(\sqrt{-1} = i\). So, \(\sqrt{-15} = \sqrt{15} * i\). Now, you have to square the entire term. The rule for squaring is to square both the 'real' and 'imaginary' parts separately, which results in \((\sqrt{15} * i)^2 = (\sqrt{15})^2 * i^2) = 15 * -1 = -15\)
03

Writing the Result in Standard Form

The standard form for a complex number is \(a + bi\). Here 'a' is the real part and 'bi' is the imaginary part. As our result is -15, it's actually a real number (not complex) despite originating from the square of an imaginary number. Therefore, the standard form is -15 + 0i, but as adding 0i doesn't change the number, you could just say -15.

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Most popular questions from this chapter

Determine whether the statement is true or false. Justify your answer. $$i^{44}+i^{150}-i^{74}-i^{109}+i^{61}=-1$$

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