Domanda del libro di testo
In Exercises 97–102, write each algebraic expression without parentheses. 1/3(3x)+[(4y)+(−4y)]
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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 102
Factor and simplify each algebraic expression.
Guida verificata passo dopo passo1
Identify the common factors in the expression: the terms involve powers of \( (4x+3) \), specifically \( (4x+3)^{-2} \) and \( (4x+3)^{-1} \).
Rewrite the expression to clearly see the common factor: \( -8(4x+3)^{-2} + 10(5x+1)(4x+3)^{-1} \). Notice that \( (4x+3)^{-2} = \frac{1}{(4x+3)^2} \) and \( (4x+3)^{-1} = \frac{1}{4x+3} \).
Factor out the smallest power of \( (4x+3) \), which is \( (4x+3)^{-2} \), from both terms: \( (4x+3)^{-2} \) times the remaining expression inside parentheses.
Inside the parentheses, divide each original term by \( (4x+3)^{-2} \): the first term becomes \( -8 \), and the second term becomes \( 10(5x+1)(4x+3)^{-1 + 2} = 10(5x+1)(4x+3)^1 = 10(5x+1)(4x+3) \).
Write the factored expression as \( (4x+3)^{-2} \left[ -8 + 10(5x+1)(4x+3) \right] \). Then, simplify inside the brackets by expanding and combining like terms if possible.
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Negative Exponents
Negative exponents indicate the reciprocal of the base raised to the positive exponent. For example, a^(-n) = 1/a^n. Understanding this helps in rewriting expressions like (4x+3)^-2 as 1/(4x+3)^2, which is essential for simplifying the given expression.
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6:37
Zero and Negative Rules
Factoring Algebraic Expressions
Factoring involves rewriting an expression as a product of its factors. Recognizing common factors, such as (4x+3)^-1 in this problem, allows you to simplify complex expressions by factoring out shared terms, making the expression easier to combine and reduce.
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05:09Introduction to Algebraic Expressions
Combining Like Terms with Rational Expressions
When expressions involve terms with variables raised to powers or negative exponents, combining like terms requires expressing them with a common denominator or base. This skill is crucial for adding or subtracting terms like those in the problem, enabling simplification into a single, reduced expression.
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05:56Introduction to Rational Equations
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