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Chapter 8: Problem 132

Factor out the specified factor. $$ 7 x^{-3 n} \text { from } 21 x^{6 n}+7 x^{3 n}+14 $$

Short Answer

Expert verified
The factored expression is \(7x^{-3n} (3x^{9n} + x^{6n} + 2x^{3n})\).

Step by step solution

01

Identify the Greatest Common Factor (GCF)

Examine the terms of the expression to identify the greatest common factor that can be factored out. We are looking to factor out \(7x^{-3n}\). The terms provided are \(21x^{6n}, 7x^{3n},\) and \(14\).
02

Rewrite Each Term

Divide each term by the factor \(7x^{-3n}\) and express them in terms of this factor: 1. \(\frac{21x^{6n}}{7x^{-3n}} = 3x^{9n}\) 2. \(\frac{7x^{3n}}{7x^{-3n}} = x^{6n}\) 3. \(\frac{14}{7x^{-3n}} = 2x^{3n}\)
03

Express the Expression Using Factored Form

Now, rewrite the expression by factoring out \(7x^{-3n}\) and compiling the results from Step 2:\[7x^{-3n} (3x^{9n} + x^{6n} + 2x^{3n})\]
04

Simplify the Expression

There is no further simplification needed since the terms resulting from factoring are distinct. Therefore, this will be the final factored expression.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Greatest Common Factor (GCF)
The greatest common factor, or GCF, is an essential concept in algebra that simplifies the process of factoring expressions. The GCF is the largest factor that all terms in an expression share. By identifying the GCF, we can simplify expressions, making them easier to manipulate or solve.
  • Start by listing all factors of each term.
  • Compare these factors to find the greatest number that evenly divides all the terms.
In this exercise, we need to factor out the GCF, which is given as \(7x^{-3n}\).
The terms \(21x^{6n}, 7x^{3n},\) and \(14\) share common factors that include both numbers and variable components. Factoring out this GCF will break down the expression into simpler parts, highlighting the underlying structure and making it more manageable.
Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and operations like addition and multiplication. They form the building blocks of algebra, allowing us to represent mathematical relationships symbolically.
  • Terms are components of expressions that are added or subtracted.
  • Each term can have coefficients (numerical factors) and variables raised to exponents.
In the given problem, the algebraic expression consists of three terms: \(21x^{6n}, 7x^{3n},\) and \(14\).
To simplify or manipulate algebraic expressions, we often need to factor them. This means breaking them down into products of simpler expressions, using methods such as finding the GCF or applying specific factoring formulas.
Exponents in Algebra
Exponents are a fundamental part of algebra that indicate how many times a base number or variable is multiplied by itself. Understanding how to work with exponents is crucial when dealing with algebraic expressions, especially in simplifying and factoring them.
  • An exponent is represented as a small number to the upper right of the base.
  • They follow specific rules for operations, such as multiplication and division.
In the exercise, we use exponents to express variables like \(x^{-3n}\), indicating repeated division.
When factoring, we divide the exponents according to these rules. For example, dividing \(x^{6n}\) by \(x^{-3n}\) involves subtracting the exponents: \(6n - (-3n) = 9n\).
Mastering the properties of exponents allows us to factor expressions efficiently and avoid errors in calculations.

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