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Ch. R - Review of Basic Concepts
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. R - Review of Basic ConceptsProblem 129
Chapter 1, Problem 129

Factor out the least power of the variable or variable expression. Assume all variables represent positive real numbers. See Example 8. 4x(2x+3)-5/9+6x2(2x+3)4/9-8x3(2x+3)13/9

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Identify the variable expressions and their exponents in each term. The terms are: \(4x(2x+3)^{-\frac{5}{9}}\), \(6x^{2}(2x+3)^{\frac{4}{9}}\), and \(-8x^{3}(2x+3)^{\frac{13}{9}}\).
Determine the least power of \(x\) among the terms. The powers of \(x\) are 1, 2, and 3 respectively, so the least power is \(x^{1}\).
Determine the least power of the expression \((2x+3)\) among the terms. The powers are \(-\frac{5}{9}\), \(\frac{4}{9}\), and \(\frac{13}{9}\) respectively, so the least power is \((2x+3)^{-\frac{5}{9}}\).
Factor out the least powers identified: \(x^{1}\) and \((2x+3)^{-\frac{5}{9}}\) from each term. This means rewriting each term as a product of the factored out expression and the remaining factors.
Write the expression as the product of the factored out terms and a sum of the simplified remaining terms inside parentheses.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Factoring Out the Least Power

Factoring out the least power involves identifying the smallest exponent of each variable or expression common to all terms and extracting it as a factor. This simplifies the expression by reducing the powers inside the parentheses, making it easier to work with or combine like terms.
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Properties of Exponents

Understanding exponent rules is essential, especially when dealing with fractional and negative exponents. Key properties include subtracting exponents when factoring out common terms and knowing how to handle expressions like (2x+3) raised to fractional powers.
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Rational Exponents

Assumption of Positive Variables

Assuming all variables represent positive real numbers allows simplification without considering absolute values or sign changes. This assumption ensures that expressions with fractional or negative exponents are well-defined and that factoring steps are valid.
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Related Practice
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Textbook Question

Factor out the least power of the variable or variable expression. Assume all variables represent positive real numbers. See Example 8. 6y3(4y−1)−3/7−8y2(4y−1)4/7+16y(4y−1)11/76y^3(4y - 1)^{-3/7} - 8y^2(4y - 1)^{4/7} + 16y(4y - 1)^{11/7}