Problem 20 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 11-46: Simplify. $$ \left(x^{2} y z^{3}\right)\left(-2 x z^{2}\right)\left(x^{3} y^{-2}\right) $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{-2x^6z^5}{y} \).

Step 1: Distribute Multiplication Across Terms

The expression given is \( (x^2 y z^3)(-2 x z^2)(x^3 y^{-2}) \). We need to multiply all the like terms together. Start by grouping all the \( x \), \( y \), and \( z \) terms from each part of the expression.

Step 2: Multiply the Coefficients

In this expression, the only coefficient we see is -2. Since it's the only number, it will remain -2 after multiplying through.

Step 3: Simplify the \( x \) Terms

Multiply the \( x \) terms: \( x^2 \), \( x \), and \( x^3 \) together. Use the rule for exponents \( a^m \times a^n = a^{m+n} \).\[ x^2 \times x \times x^3 = x^{2+1+3} = x^6 \]

Step 4: Simplify the \( y \) Terms

For \( y \) terms: \( y \) and \( y^{-2} \). When you have an exponent with a zero value, it equals 1.\[ y \times y^{-2} = y^{1+(-2)} = y^{-1} \]

Step 5: Simplify the \( z \) Terms

Simplify the \( z \) terms: \( z^3 \) and \( z^2 \) using the same exponent rule.\[ z^3 \times z^2 = z^{3+2} = z^5 \]

Step 6: Write the Simplified Expression

Combine all the simplified terms together with the multiplied coefficient to get the final expression:\[ -2x^6 y^{-1} z^5 \]

Step 7: Final Concise Representation

Finally, to represent \( y^{-1} \) with positive exponents, write it in the denominator:\[ \frac{-2x^6z^5}{y} \]

Key Concepts

Polynomial Expressions

Polynomial expressions are a fundamental component of algebra, consisting of variables, coefficients, and operations like addition, subtraction, and multiplication. Each part of the expression, such as a single term like \(x^2yz^3\), is called a monomial. Monomials can be added or multiplied to form more complex polynomial expressions. In our exercise, we're dealing with a multiplication of three monomials: \((x^2 y z^3)(-2 x z^2)(x^3 y^{-2})\).
Understanding polynomials involves recognizing these terms' structure and how they interact. Each term is composed of:

• **Variables**: Represent quantities that can change or vary within the expression, such as \(x\), \(y\), and \(z\).
• **Coefficients**: Numerical factors that determine the multiple of the variable, such as the -2 in the second term.
• **Exponents**: Small numbers written above and to the right of a variable, indicating how many times the variable is used as a factor.

The challenge is to simplify the expression, which involves consolidating these components systematically using algebraic rules.

Exponent Rules

Exponent rules are essential when simplifying expressions with powers. One of the most important rules is the **product of powers rule**, which states that when you multiply two exponents with the same base, you add their exponents: \(a^m \times a^n = a^{m+n}\).
In our expression, we used these rules multiple times:

• For the \(x\) terms: \(x^2 \times x \times x^3 = x^{2+1+3} = x^6\).
• For the \(y\) terms: \(y \times y^{-2} = y^{1+(-2)} = y^{-1}\).
• For the \(z\) terms: \(z^3 \times z^2 = z^{3+2} = z^5\).

Another rule is the **negative exponent rule**, which transforms a term with a negative exponent into its reciprocal, allowing us to work with positive exponents: \(a^{-n} = \frac{1}{a^n}\).
In our final expression, \(y^{-1}\) becomes \(\frac{1}{y}\). Learning and applying these rules makes it easier to simplify polynomial expressions correctly.

Multiplying Monomials

Multiplying monomials involves using both the structure of the expression and exponent rules to combine similar terms. The exercise exemplifies this by requiring us to systematically multiply terms like \((x^2 y z^3)(-2 x z^2)(x^3 y^{-2})\):

• **Step 1: Group like terms**: Collect all \(x\), \(y\), and \(z\) terms separately to simplify the problem.
• **Step 2: Multiply constants**: In our problem, -2 is the only constant, so it remains -2.
• **Step 3: Apply exponent rules**: Use the product of powers rule to combine terms with the same base by adding their exponents.

Ultimately, the simplification reduces the complexity of the expression and consolidates all information into a single, cohesive expression: \(\frac{-2x^6z^5}{y}\). Getting comfortable with multiplying monomials prepares students for more complex algebraic operations and lays a groundwork for further mathematical learning.