Domanda del libro di testo
In Exercises 53–64, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. [2(cos 80° + i sin 80°)]³
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Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Blitzer 3rd EditionCh. 5 - Complex Numbers, Polar Coordinates and Parametric EquationsProblema 5.1.51
Blitzer 3rd EditionCh. 5 - Complex Numbers, Polar Coordinates and Parametric EquationsProblema 5.1.51Capitolo 5, Problema 5.1.51
In Exercises 37–52, perform the indicated operations and write the result in standard form.
(3√(−5) )( −4√(−12) )
Guida verificata passo dopo passo1
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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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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Percorso guidato
04:47
Complex Numbers In Polar Form
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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Percorso guidato
5:02Multiplying 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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Percorso guidato
2:20Imaginary Roots with the Square Root Property
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Domanda del libro di testo
In Exercises 45–52, find the quotient z₁/z₂ of the complex numbers. Leave answers in polar form. In Exercises 49–50, express the argument as an angle between 0° and 360°.
z₁ = cos 80° + i sin 80°
z₂ = cos 200° + i sin 200°
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Domanda del libro di testo
In Exercises 49–58, convert each rectangular equation to a polar equation that expresses r in terms of θ.
x² = 6y
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Domanda del libro di testo
In Exercises 37–52, perform the indicated operations and write the result in standard form.
√(−8) (√(−3) − √5 )
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Domanda del libro di testo
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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Domanda del libro di testo
In Exercises 65–68, find all the complex roots. Write roots in polar form with θ in degrees. The complex square roots of 9(cos 30° + i sin 30°)
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