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Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations
Blitzer - Trigonometry 3rd Edition
Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook
All textbooksBlitzer 3rd EditionCh. 5 - Complex Numbers, Polar Coordinates and Parametric EquationsProblem 5.1.51
Chapter 5, Problem 5.1.51

In Exercises 37–52, perform the indicated operations and write the result in standard form.
(3√(−5) )( −4√(−12) )

Verified step by step guidance
1
Recognize that the expression involves multiplying two complex numbers in the form of \( a\sqrt{-1} \), which can be rewritten using the imaginary unit \( i = \sqrt{-1} \). So, rewrite \( \sqrt{-5} \) as \( \sqrt{5}i \) and \( \sqrt{-12} \) as \( \sqrt{12}i \).
Rewrite the original expression \( (3\sqrt{-5})(-4\sqrt{-12}) \) as \( (3\sqrt{5}i)(-4\sqrt{12}i) \).
Multiply the coefficients and the square root parts separately: multiply \( 3 \times -4 \) and \( \sqrt{5} \times \sqrt{12} \), and multiply the imaginary units \( i \times i \).
Recall that \( i \times i = i^2 = -1 \), so replace \( i^2 \) with \( -1 \) in your expression.
Simplify the expression by multiplying all numerical parts and applying the \( -1 \) from \( i^2 \), then write the final result in the form \( a + bi \), which is the standard form for complex numbers.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Complex Numbers and Standard Form

Complex numbers are expressed in the form a + bi, where a is the real part and b is the imaginary part. The standard form requires separating and simplifying these parts after performing operations like multiplication.
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Multiplication of Complex Numbers

Multiplying complex numbers involves using the distributive property (FOIL) and applying i² = -1 to simplify terms. This process combines real and imaginary parts to form a new complex number.
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Multiplying Complex Numbers

Simplifying Square Roots of Negative Numbers

Square roots of negative numbers are expressed using imaginary unit i, where √(-n) = i√n. Simplifying these roots is essential before performing multiplication to correctly handle the imaginary components.
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Imaginary Roots with the Square Root Property
Related Practice
Textbook Question

In Exercises 41–43, eliminate the parameter. Write the resulting equation in standard form.


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In Exercises 53–58, perform the indicated operation(s) and write the result in standard form. (2 + i)² − (3 − i)²

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Textbook Question

In Exercises 1–8, parametric equations and a value for the parameter t are given. Find the coordinates of the point on the plane curve described by the parametric equations corresponding to the given value of t. x = 4 + 2 cos t, y = 3 + 5 sin t; t = π/2