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Ch. P - Fundamental Concepts of Algebra
Blitzer - College Algebra 8th Edition
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Tutti i libri di testoBlitzer 8th EditionCh. P - Fundamental Concepts of AlgebraProblema 117
Capitolo 1, Problema 117

Add or subtract as indicated. 12/25x^2+11/15x^3

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Step 1: Identify the terms in the expression. The given expression is \( \frac{12}{25}x^2 + \frac{11}{15}x^3 \). Notice that the terms have different powers of \(x\), so they are not like terms and cannot be directly added or subtracted.
Step 2: Factor out the greatest common factor (GCF) if possible. In this case, the coefficients \( \frac{12}{25} \) and \( \frac{11}{15} \) do not share a common factor, and the powers of \(x\) are different, so no factoring is possible.
Step 3: Rewrite the expression as it is already simplified. Since the terms are not like terms, the expression remains \( \frac{12}{25}x^2 + \frac{11}{15}x^3 \).
Step 4: If required, you can express the terms with a common denominator for the coefficients. The least common denominator (LCD) of \(25\) and \(15\) is \(75\). Rewrite each fraction with a denominator of \(75\): \( \frac{12}{25} = \frac{36}{75} \) and \( \frac{11}{15} = \frac{55}{75} \). The expression becomes \( \frac{36}{75}x^2 + \frac{55}{75}x^3 \).
Step 5: Combine the rewritten terms if necessary. Since the terms still have different powers of \(x\), they cannot be combined further. The final simplified expression is \( \frac{36}{75}x^2 + \frac{55}{75}x^3 \).

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Rational Expressions

Rational expressions are fractions where the numerator and the denominator are polynomials. To add or subtract these expressions, it is essential to have a common denominator. This involves finding the least common multiple (LCM) of the denominators, which allows for the combination of the fractions into a single expression.
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Rationalizing Denominators

Finding a Common Denominator

Finding a common denominator is a crucial step in adding or subtracting rational expressions. The common denominator must be the least common multiple of the individual denominators. Once identified, each fraction is adjusted by multiplying its numerator and denominator by the necessary factors to achieve this common denominator, enabling the fractions to be combined.
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Rationalizing Denominators

Combining Like Terms

Combining like terms involves simplifying expressions by adding or subtracting coefficients of terms that have the same variable raised to the same power. After obtaining a common denominator and combining the numerators, it is important to simplify the resulting expression by merging any like terms, which streamlines the final answer and makes it easier to interpret.
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Combinations
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