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Chapter 2: Problem 84

Simplify the complex number and write it in standard form.$$(-i)^{3}$$

Short Answer

Expert verified
The simplified form of the complex number (-i)^3 is i.

Step by step solution

01

Simplify the Power of i

The imaginary unit 'i' has specific properties when raised to different powers. More specifically it repeats every 4th power. Considering the given problem, we have (-i)^3. Simplify this as follows using these properties (-i) * (-i) * (-i) = -i * -i * -i = -1 * -i = i.
02

Rewrite Result in Standard Form

Complex numbers are written in the form a + bi. Here, 'a' is the real part and 'b' is the imaginary part. Rewrite the simplified complex number in this form. However, since there is only imaginary part and no real part, the simplified complex number is as it was found in step one, i.e., i.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Imaginary Unit
The imaginary unit, denoted as \( i \), is a fundamental concept in complex numbers. It is defined by the property \( i^2 = -1 \). This means that \( i \) is the square root of -1, a number that cannot be placed on the traditional real number line. Understanding \( i \) is essential, as it allows for the expansion of mathematics into the complex plane. Complex numbers, which include the imaginary unit, are useful in various advanced fields such as engineering and physics.
Powers of i
The powers of the imaginary unit \( i \) exhibit a unique cyclical pattern every four powers. This cyclical nature makes calculations with powers of \( i \) more manageable. Here's how it works:
  • \( i^1 = i \)
  • \( i^2 = -1 \)
  • \( i^3 = -i \)
  • \( i^4 = 1 \)
After \( i^4 \), the powers repeat the cycle: \( i^5 = i \), \( i^6 = -1 \), \( i^7 = -i \), and \( i^8 = 1 \), continuing in this loop. This cycle simplifies computations, such as with \((-i)^3\). Recognizing the pattern, we have \((-i)^3 = -i \), which equates to the power of \( i^3 \). This repetition reduces the need for laborious calculation.
Standard Form of Complex Numbers
Complex numbers are usually expressed in what is known as the "standard form," written as \( a + bi \). In this expression:
  • \( a \) represents the real part of the complex number.
  • \( b \) represents the coefficient of the imaginary part, i.e., \( bi \).
For example, in the calculation of \((-i)^3\), the simplified result was \( i \). Since there is no real part (real part \( a = 0 \)), it can be written as \( 0 + i \), though it is often left simply as \( i \) for simplicity. This standard form helps in identifying and separating real and imaginary parts clearly, aiding in operations like addition, subtraction, and multiplication of complex numbers.

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Most popular questions from this chapter

Find two positive real numbers whose product is a maximum. The sum of the first and three times the second is 42.

Write the standard form of the equation of the parabola that has the indicated vertex and passes through the given point. Vertex: (5,12)\(;\) point: (7,15)

Write the quadratic function in standard form and sketch its graph. Identify the vertex, axis of symmetry, and \(x\) -intercept(s). $$f(x)=-x^{2}-4 x+1$$

Use Descartes's Rule of Signs to determine the possible numbers of positive and negative real zeros of the function. $$f(x)=-5 x^{3}+x^{2}-x+5$$

Use a graphing utility to graph the quadratic function. Find the \(x\) -intercept(s) of the graph and compare them with the solutions of the corresponding quadratic equation when \(f(x)=0\) $$f(x)=\frac{7}{10}\left(x^{2}+12 x-45\right)$$
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