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Chapter 2: Problem 83

Simplify the complex number and write it in standard form.$$-14 i^{5}$$.

Short Answer

Expert verified
The simplified form of the complex number \(-14i^5\) is \(-14i\).

Step by step solution

01

Breaking down the exponent

Express the exponent of \(i\) as \(5 = 4 + 1\). Thus \(-14i^5\) can be written as \(-14(i^4 \times i)\).
02

Simplify using the property of \(i\)

Since \(i^4 = 1\), \(-14(i^4 \times i)\) simplifies to \(-14 \times 1 \times i = -14i\)

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Key Concepts

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

Standard Form
In mathematics, the standard form of a complex number is an easy-to-read way of expressing it. It is written as \(a + bi\), where \(a\) and \(b\) are real numbers. In this representation:
  • \(a\) is the real part.
  • \(b\) is the imaginary part.

This form is used to simplify the addition, subtraction, multiplication, and division of complex numbers. When given a complex number like \(-14i^5\), our task is to simplify it to its standard form.
By simplifying \(-14i^5\) to \(-14i\), the number ends up directly in its standard form. The result \(-14i\) shows that the real part is 0, and the imaginary part is \(-14\).
Imaginary Unit
The concept of the imaginary unit, represented by \(i\), is central to complex numbers. The imaginary unit is defined as \(i = \sqrt{-1}\), which means \(i^2 = -1\).
This property is crucial when simplifying expressions involving powers of \(i\).When we raise \(i\) to higher powers, we observe a repeating cycle in the values:
  • \(i^1 = i\)
  • \(i^2 = -1\)
  • \(i^3 = -i\)
  • \(i^4 = 1\)
After this, the cycle repeats every four powers. Thus, knowing this cycle helps simplify any power of \(i\). In our example, \(i^5 \) can be broken down as \(i^4 \times i\), resulting in \(1 \times i = i\).
Exponents
Exponents play a pivotal role when working with complex numbers that we aim to express in standard form, particularly those involving powers of the imaginary unit \(i\).
Understanding the cycle of powers of \(i\) means we can simplify larger exponents with ease.For example, higher powers can be rewritten by considering multiples of four, due to the periodicity of \(i\):
  • \(i^n\) where \(n = 4k \Rightarrow i^n = 1\)
  • \(i^n\) where \(n = 4k + 1 \Rightarrow i^n = i\)
  • \(i^n\) where \(n = 4k + 2 \Rightarrow i^n = -1\)
  • \(i^n\) where \(n = 4k + 3 \Rightarrow i^n = -i\)

In simplifying \(-14i^5\), identifying \(5\) as \(4 + 1\) helps recognize \(i^5\) evaluates to \(i\), as explained in the pattern. This simplification allows us to express the complex number easily in standard form as \(-14i\).

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Most popular questions from this chapter

Write the quadratic function in standard form and sketch its graph. Identify the vertex, axis of symmetry, and \(x\) -intercept(s). $$f(x)=-\frac{1}{3} x^{2}+3 x-6$$

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