Textbook Question
In Exercises 1–6, find all numbers that must be excluded from the domain of each rational expression. (x−1)/(x2+11x+10)
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Ch. P - Fundamental Concepts of Algebra
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- 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
Chapter 1, Problem 6
Evaluate each expression in Exercises 1–12, or indicate that the root is not a real number. √−25
Verified step by step guidance1
Recognize that the problem involves taking the square root of a negative number, specifically √−25.
Recall that the square root of a negative number is not a real number in the set of real numbers. Instead, it is expressed using imaginary numbers.
Rewrite the square root of the negative number using the imaginary unit 'i', where i = √−1. For this problem, √−25 can be rewritten as √(25) × √(−1).
Simplify the square root of 25 to 5, and the square root of −1 to 'i'. Combine these results to express the solution as 5i.
Conclude that the root is not a real number, but rather an imaginary number, specifically 5i.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Square Roots
A square root of a number 'x' is a value 'y' such that y² = x. For real numbers, square roots are defined only for non-negative values. When evaluating square roots, if 'x' is negative, the result is not a real number, leading to the concept of imaginary numbers.
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Imaginary Roots with the Square Root Property
Imaginary Numbers
Imaginary numbers are defined as multiples of the imaginary unit 'i', where i = √−1. This concept extends the real number system to include solutions to equations that do not have real solutions, such as the square root of negative numbers. For example, √−25 can be expressed as 5i.
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05:02Square Roots of Negative Numbers
Complex Numbers
Complex numbers are numbers that have a real part and an imaginary part, expressed in the form a + bi, where 'a' is the real part and 'b' is the coefficient of the imaginary part. This concept is essential for understanding how to work with square roots of negative numbers and performing operations involving both real and imaginary components.
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Related Practice
Textbook Question
In Exercises 1–16, evaluate each algebraic expression for the given value or values of the variable(s). x2-6x+3, for x=7
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In Exercises 1–16, evaluate each algebraic expression for the given value or values of the variable(s). x^2+3x, for x=8
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Let A = {a, b, c}, B = {a, c, d, e}, and C = {a, d, f, g}. Find the indicated set A ∪ B.
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Textbook Question
Evaluate each exponential expression in Exercises 1–22. (−3)0
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Textbook Question
In Exercises 7–14, simplify each rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression. (3x−9)/(x2−6x+9)
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