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In Exercises 83–90, perform the indicated operation or operations. (3x+4y)2−(3x−4y)2
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Ch. P - Fundamental Concepts of Algebra
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 1, Problema 83
State the name of the property illustrated. 1/(x+3) (x+3)=1, x≠−3
Guida verificata passo dopo passo1
Identify the expression given: \(\frac{1}{x+3} \cdot (x+3) = 1\), with the restriction \(x \neq -3\) to avoid division by zero.
Recognize that multiplying a number by its reciprocal results in 1. Here, \(\frac{1}{x+3}\) is the reciprocal of \((x+3)\).
This illustrates the property that any nonzero number multiplied by its multiplicative inverse (reciprocal) equals 1.
State the property name: This is the Multiplicative Inverse Property.
Note the importance of the restriction \(x \neq -3\) to ensure the denominator is not zero, which is necessary for the property to hold.
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Multiplicative Inverse Property
This property states that for any nonzero number a, multiplying it by its reciprocal (1/a) results in 1. In the given expression, (1/(x+3)) multiplied by (x+3) equals 1, illustrating that (x+3) and its reciprocal are multiplicative inverses, provided x ≠ -3 to avoid division by zero.
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Percorso guidato
5:36
Change of Base Property
Domain Restrictions
Domain restrictions specify values that variables cannot take to avoid undefined expressions. Here, x ≠ -3 because substituting -3 would make the denominator zero, which is undefined in algebra. Recognizing these restrictions is essential to correctly apply properties and solve equations.
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Percorso guidato
3:51Domain Restrictions of Composed Functions
Properties of Rational Expressions
Rational expressions are fractions involving polynomials. Understanding how to simplify, multiply, and identify restrictions in rational expressions is key. The problem demonstrates multiplying a rational expression by its denominator, simplifying to 1, which relies on these properties.
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Percorso guidato
02:58Rationalizing Denominators
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