Domanda del libro di testo
Solve each equation in Exercises 47–64 by completing the square.
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Ch. 1 - Equations and Inequalities
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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TOC.tableOfContents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Capitolo 2, Problema 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 problem involves multiplying two expressions with square roots of negative numbers. Recall that the square root of a negative number can be expressed using the imaginary unit i, where i = √-1.
Rewrite each square root in terms of i. For example, √-5 can be written as i√5, and √-12 can be written as i√12.
Substitute these rewritten forms into the expression: (3√-5)(-4√-12) becomes (3i√5)(-4i√12).
Simplify the multiplication step by step. Multiply the coefficients (3 and -4), the imaginary units (i and i), and the square root terms (√5 and √12). Remember that i * i = -1.
Combine the results into a single expression. Simplify the square root multiplication (√5 * √12 = √60) and reduce √60 to its simplest radical form if possible. Write the final result in standard form as a complex number (a + bi).
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Complex Numbers
Complex numbers are numbers that have a real part and an imaginary part, typically expressed in the form a + bi, where 'a' is the real part and 'b' is the coefficient of the imaginary unit 'i', which is defined as the square root of -1. Understanding complex numbers is essential for performing operations involving square roots of negative numbers, as they allow us to express these roots in a meaningful way.
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Dividing Complex Numbers
Multiplication of Complex Numbers
When multiplying complex numbers, the distributive property is applied, similar to multiplying polynomials. For example, when multiplying (a + bi)(c + di), the result is ac + adi + bci + bdi². Since i² = -1, this simplifies to (ac - bd) + (ad + bc)i. Mastery of this process is crucial for correctly performing operations on complex numbers.
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05:02Multiplying Complex Numbers
Standard Form of Complex Numbers
The standard form of a complex number is expressed as a + bi, where 'a' and 'b' are real numbers. This format is important because it clearly separates the real and imaginary components, making it easier to interpret and manipulate complex numbers. When performing operations, the final result should be presented in this standard form for clarity and consistency.
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05:02Multiplying Complex Numbers
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