Domanda del libro di testo
In Exercises 27–32, select the representations that do not change the location of the given point. (−2, 7π/6) (−2, −5π/6)
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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 31
Blitzer 3rd EditionCh. 5 - Complex Numbers, Polar Coordinates and Parametric EquationsProblema 31Capitolo 5, Problema 31
In Exercises 29–36, simplify and write the result in standard form. ____ √−108
Guida verificata passo dopo passo1
Recognize that the expression involves the square root of a negative number, which means the result will be a complex number. Recall that \(\sqrt{-a} = \sqrt{a} \times i\), where \(i\) is the imaginary unit with the property \(i^2 = -1\).
Rewrite the expression \(\sqrt{-108}\) as \(\sqrt{108} \times i\) to separate the imaginary unit from the real number under the root.
Simplify \(\sqrt{108}\) by factoring 108 into its prime factors or perfect squares. For example, \(108 = 36 \times 3\), and since \(\sqrt{36} = 6\), you can write \(\sqrt{108} = \sqrt{36 \times 3} = 6\sqrt{3}\).
Substitute back to get the expression in terms of \(i\): \(\sqrt{-108} = 6\sqrt{3} \times i\).
Write the final answer in standard form for complex numbers, which is $a + bi$. Since there is no real part here, the expression is \(0 + 6\sqrt{3}i\).
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Simplifying Square Roots of Negative Numbers
When simplifying the square root of a negative number, recognize that it involves imaginary numbers. The square root of a negative number can be expressed as the product of the imaginary unit 'i' (where i² = -1) and the square root of the corresponding positive number.
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Percorso guidato
2:20
Imaginary Roots with the Square Root Property
Prime Factorization for Simplifying Radicals
To simplify a square root, break down the number inside the root into its prime factors. Pair factors to extract them from under the root, simplifying the expression. For example, √108 can be factored into √(36 × 3) = 6√3.
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Percorso guidato
6:08Factoring
Standard Form of Complex Numbers
The standard form of a complex number is a + bi, where 'a' is the real part and 'b' is the imaginary coefficient. After simplifying the radical, express the result in this form to clearly separate real and imaginary components.
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Percorso guidato
04:47Complex Numbers In Polar Form
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Domanda del libro di testo
In Exercises 29–36, simplify and write the result in standard form.
√−196
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Domanda del libro di testo
In Exercises 27–36, write each complex number in rectangular form. If necessary, round to the nearest tenth. 8(cos 7π/4 + i sin 7π/4)
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Domanda del libro di testo
In Exercises 30–31, find all the complex roots. Write roots in polar form with θ in degrees. The complex cube roots of 125(cos 165° + i sin 165°)
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Domanda del libro di testo
In Exercises 13–34, test for symmetry and then graph each polar equation. r = 1 − 3 sin θ
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Domanda del libro di testo
In Exercises 21–40, eliminate the parameter t. Then use the rectangular equation to sketch the plane curve represented by the given parametric equations. Use arrows to show the orientation of the curve corresponding to increasing values of t. (If an interval for t is not specified, assume that −∞ < t < ∞. x = 1 + 3 cos t, y = 2 + 3 sin t; 0 ≤ t < 2π
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