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Chapter 8: Problem 5

Simplify each expression to a single complex number. $$ \frac{2+\sqrt{-12}}{2} $$

Short Answer

Expert verified
The expression simplifies to \(1 + i\sqrt{3}\).

Step by step solution

01

Identify the Imaginary Component

The expression under the square root, \(-12\), is a negative number, which introduces an imaginary component. We rewrite \(\sqrt{-12}\) as \(\sqrt{-1 \times 12}\), which results in \(i\sqrt{12}\). Remember, \(i\) represents the imaginary unit where \(i^2 = -1\).
02

Simplify the Square Root

Simplify \(\sqrt{12}\). This can be simplified to \(\sqrt{4 \times 3}\), which results in \(\sqrt{4} \times \sqrt{3} = 2\sqrt{3}\). Therefore, \(\sqrt{-12} = i \times 2\sqrt{3} = 2i\sqrt{3}\).
03

Rewrite the Initial Expression

Substitute \(\sqrt{-12}\) with \(2i\sqrt{3}\) in the original expression. We have: \[\frac{2 + 2i\sqrt{3}}{2}\]
04

Simplify the Fraction

Distribute the division by 2 to both terms in the numerator: \[\frac{2}{2} + \frac{2i\sqrt{3}}{2}\]This simplifies to: \[1 + i\sqrt{3}\]The expression is now simplified to a single complex number.

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Key Concepts

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

Imaginary Unit
In mathematics, complex numbers are numbers that include both a real part and an imaginary part. The imaginary part is represented using the imaginary unit, denoted by the letter \(i\). This imaginary unit is defined by the property \(i^2 = -1\). It is used to represent the square root of negative numbers. This is essential because the square root of any negative number introduces complexity outside the realm of real numbers.
  • Whenever you encounter a square root of a negative number, you can rewrite it in terms of \(i\).
  • For example, \(\sqrt{-12}\) can be rewritten as \(\sqrt{-1 \times 12} = \sqrt{-1} \times \sqrt{12} = i \sqrt{12}\).
  • This is a fundamental step in handling negative roots, as it separates the imaginary component from the real component.
Square Root Simplification
Square root simplification is a useful technique when simplifying expressions involving radical terms. The goal is to express the square root in its simplest form, often by factoring the number under the square root to find perfect squares.
  • Consider \(\sqrt{12}\); it can be broken down into \(\sqrt{4 \times 3}\).
  • The square root of a product is the product of the roots: \(\sqrt{4} \times \sqrt{3} = 2\sqrt{3}\), since \(\sqrt{4} = 2\).
  • So, \(\sqrt{-12} = i \times 2\sqrt{3} = 2i\sqrt{3}\) after including the imaginary unit \(i\).
This process reduces the complexity of the expression, making further calculations simpler.
Fraction Simplification
Fraction simplification involves reducing fractions to their simplest form. This is achieved by dividing both the numerator and the denominator by their greatest common divisor (GCD). In complex numbers, this principle still applies.
  • For the expression \(\frac{2 + 2i\sqrt{3}}{2}\), distribute the division into each part of the numerator.
  • This results in \(\frac{2}{2} + \frac{2i\sqrt{3}}{2}\).
  • Simplify each term: \(\frac{2}{2} = 1\) and \(\frac{2i\sqrt{3}}{2} = i\sqrt{3}\).
  • Thus, the expression simplifies to \(1 + i\sqrt{3}\), a complex number in its simplest form.
This final expression effectively represents the same quantity as the original, but in a more concise form.

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Most popular questions from this chapter

Compute each of the following, leaving the result in polar \(r e^{i \theta}\) form. $$ \sqrt{9 e^{\frac{3 \pi}{2}}} $$

Rewrite each complex number into polar \(r e^{1 \theta}\) form. $$ 4+7 i $$

Sketch a graph of the polar equation. $$ r=3 \cos (\theta) $$

Rewrite each complex number into polar \(r e^{1 \theta}\) form. $$ -4 i $$

Sketch the parametric equations for \(-2 \leq t \leq 2\). $$ \left\\{\begin{array}{l} x(t)=2 t-2 \\ y(t)=t^{3} \end{array}\right. $$
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