Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations

Blitzer3rd EditionTrigonometryISBN: 9780137316601Non è quello che usi tu?Cambia libro di testo

Blitzer 3rd Edition

Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations

Problema 45

Capitolo 5, Problema 45

In Exercises 37–52, perform the indicated operations and write the result in standard form. ___ −8 + √−32 / 24

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Identify the expression to simplify: $-8 + \frac{\sqrt{-32}}{24}$. Notice that the square root involves a negative number, which means we will use imaginary numbers.

Rewrite the square root of the negative number using the imaginary unit $i$, where $i = \sqrt{-1}$. So, $\sqrt{-32} = \sqrt{32} \times \sqrt{-1} = \sqrt{32}i$.

Simplify $\sqrt{32}$ by expressing it as $\sqrt{16 \times 2} = \sqrt{16} \times \sqrt{2} = 4\sqrt{2}$. Therefore, $\sqrt{-32} = 4\sqrt{2}i$.

Substitute back into the expression: $-8 + \frac{4\sqrt{2}i}{24}$. Simplify the fraction $\frac{4\sqrt{2}i}{24}$ by dividing numerator and denominator by 4, resulting in $\frac{\sqrt{2}i}{6}$.

Write the expression in standard form $a + bi$, where $a$ and $b$ are real numbers. Here, $a = -8$ and $b = \frac{\sqrt{2}}{6}$. So the expression is $-8 + \frac{\sqrt{2}}{6}i$.

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Complex Numbers and Imaginary Unit

Complex numbers consist of a real part and an imaginary part, expressed as a + bi, where i is the imaginary unit with the property i² = -1. Understanding how to represent and manipulate these numbers is essential when dealing with square roots of negative numbers.

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Introduction to Complex Numbers

Simplifying Square Roots of Negative Numbers

To simplify the square root of a negative number, factor out the imaginary unit i by rewriting √(-a) as i√a, where a is positive. This allows the expression to be converted into a form involving real numbers and i, facilitating further operations.

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Imaginary Roots with the Square Root Property

Operations with Complex Numbers in Standard Form

Standard form for complex numbers is a + bi, where a and b are real numbers. Performing addition, subtraction, multiplication, or division requires combining like terms and applying algebraic rules, including i² = -1, to simplify the expression correctly.

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Complex Numbers In Polar Form

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