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Chapter 5: Problem 43

Simplify. $$ (-x y z x 4 y-2 z 3)-4 $$

Short Answer

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Step by step solution

01

Identify the Expression Components

First, break down the given expression. You have \((-x y z x 4 y^{-2} z^3)-4\).This expression consists of a polynomial term and a constant term. The polynomial term is \(-x y z x 4 y^{-2} z^3\),and the constant term is \(-4\).

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Key Concepts

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

Polynomial Terms
In algebra, a polynomial term is an expression made up of variables and coefficients, which are combined using multiplication and powers. The term "polynomial" generally refers to multiple terms connected by addition or subtraction; however, a single term can also be referred to as a polynomial term. For example, when dealing with the expression \[ -x y z x 4 y^{-2} z^3 \] we can see multiple variables like \(x, y, \text{and } z\) as well as numerical coefficients such as \(4\). These components mean our expression is a polynomial term.
  • **Variables:** Symbols that represent numbers; in our case: \(x, y, \text{and } z\).
  • **Coefficients:** Numbers that multiply the variables; in the expression, it is \(4\).
  • **Powers/Exponents:** Indicate how many times a variable is multiplied by itself; for example, \(y^{-2}\) means \(y\) is in the denominator.
Recognizing these components helps you understand the structure of polynomials and is useful when simplifying or manipulating expressions.
Constant Terms
Constant terms in algebra are the numbers that stand alone without variables accompanying them. They are fixed values and don't involve any variables or change. In the expression provided \[ (-x y z x 4 y^{-2} z^3)-4 \] we notice "\(-4\)" is the constant term because it doesn't have a variable attached. Constants are crucial as they often represent offset values or fixed components in functions.
  • They provide a base or starting point in an equation, unaffected by changes in variable values.
  • In equations, constant terms are those that we compare or balance against variable-based expressions.
Understanding constant terms enables you to identify parts of an equation's structure that remain unchanged, no matter the input variable values.
Algebraic Simplification
Algebraic simplification is the process of transforming an expression into a simpler or more manageable form. This involves combining like terms, reducing coefficients, and ensuring all exponents are correctly processed. With algebraic simplification, you're essentially cleaning up expressions to make them easier to work with.When simplifying complex expressions like \[ (-x y z x 4 y^{-2} z^3)-4 \], we need to be attentive to the rules of exponents and coefficients:
  • Combine like terms: Involves adding or subtracting terms with the same variable and power.
  • Use exponent rules: Such as \(a^n \times a^m = a^{n+m}\), to manage terms with powers.
  • Assure coefficients are simple: Simplify expressions like \(4 \times 1 = 4\).
In practice, simplifying involves careful attention to detail to avoid errors. The goal is to make equations more straightforward, while keeping integrity of the original expression.

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