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 8

Capitolo 5, Problema 8

Perform the indicated operations and write the result in standard form. √−32 − √−18

Guida verificata passo dopo passo

Recognize that the square roots of negative numbers involve imaginary numbers. Recall that \(\sqrt{-a} = \sqrt{a} \times i\), where \(i = \sqrt{-1}\).

Rewrite each term using the imaginary unit \(i\): \(\sqrt{-32} = \sqrt{32} \times i\) and \(\sqrt{-18} = \sqrt{18} \times i\).

Simplify the square roots of the positive numbers under the radicals by factoring out perfect squares: \(\sqrt{32} = \sqrt{16 \times 2} = 4\sqrt{2}\) and \(\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}\).

Substitute the simplified forms back into the expression: \(4\sqrt{2}i - 3\sqrt{2}i\).

Combine like terms by factoring out \(\sqrt{2}i\): \((4 - 3)\sqrt{2}i\), which simplifies to \(1 \times \sqrt{2}i\), or simply \(\sqrt{2}i\).

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Imaginary Numbers and Complex Numbers

Imaginary numbers arise from the square roots of negative numbers, defined using the imaginary unit i, where i² = -1. Complex numbers combine real and imaginary parts in the form a + bi. Understanding this allows you to rewrite √-32 and √-18 in terms of i.

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Percorso guidato

3:31

Introduction to Complex Numbers

Simplifying Square Roots

Simplifying square roots involves factoring the radicand into perfect squares and other factors. For example, √32 can be simplified to 4√2 because 16 is a perfect square. This process helps in expressing the terms in a simpler form before performing operations.

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Percorso guidato

2:20

Imaginary Roots with the Square Root Property

Operations with Complex Numbers

Adding or subtracting complex numbers requires combining like terms: real parts with real parts and imaginary parts with imaginary parts. After simplifying the square roots, you perform the indicated subtraction by handling the imaginary components accordingly to write the result in standard form a + bi.

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Percorso guidato

4:22

Dividing Complex Numbers

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