Problem 28 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 11-46: Simplify. $$ \left(2 x^{2} y^{-5}\right)\left(6 x^{-3} y\right)\left(\frac{1}{3} x^{-1} y^{3}\right) $$

Step-by-Step Solution

Answer

\(\frac{4}{x^2 y}\)

Step 1: Distribute Constants

First, multiply all the constant coefficients together. In this case, multiply 2, 6, and \(\frac{1}{3}\) together: \(2 \times 6 \times \frac{1}{3} = 4\).

Step 2: Simplify the x terms

Now, multiply the \(x\) terms. Add the exponents: \(x^2 \times x^{-3} \times x^{-1}\). This becomes \(x^{2 + (-3) + (-1)} = x^{-2}\).

Step 3: Simplify the y terms

Next, multiply the \(y\) terms. Add the exponents: \(y^{-5} \times y^{1} \times y^{3}\). This results in \(y^{-5 + 1 + 3} = y^{-1}\).

Step 4: Combine Simplified Terms

Combine all simplified terms together: the constant from Step 1 and the simplified \(x\) and \(y\) terms from Steps 2 and 3. This gives us \(4 x^{-2} y^{-1}\).

Step 5: Write Final Answer in Positive Exponents

Since we want only positive exponents, rewrite \(4 x^{-2} y^{-1}\) as a fraction: \(\frac{4}{x^2 y}\).

Key Concepts

Simplifying Expressions

When you encounter an algebraic expression, the goal is often to reduce it to its simplest form. Simplifying expressions makes them easier to understand and work with. This means combining like terms, such as constants or similar variables, to create an expression that is as compact as possible.

In the given exercise, simplifying involves distributing constants and combining variables with exponents.

• Combine constant factors: First, multiply all separate numerical parts of the expression together. This means combining any numbers not associated with a variable.
• Add or subtract exponents: When working with variables, follow the rules for exponent arithmetic. When the same base is raised to different powers, add those exponents if multiplying, or subtract if dividing.

Simplifying is a fundamental skill in algebra that helps reduce complexity. By following these basic steps, expressions become cleaner and more manageable.

Exponents Rules

Exponents convey how many times a number, or variable, is multiplied by itself. Understanding and applying exponent rules is crucial when simplifying expressions.

Key rules used in this exercise include:

• The Product of Powers Rule: When multiplying variables with the same base, add their exponents: \(a^m \times a^n = a^{m+n}\).
• The Quotient of Powers Rule: When dividing variables with the same base, subtract the exponents: \(\frac{a^m}{a^n} = a^{m-n}\).
• Negative Exponents Rule: A negative exponent means that the base is on the wrong side of the fraction line; it indicates reciprocation: \(a^{-n} = \frac{1}{a^n}\).

In the step-by-step solution, these rules help transform the expression \(x^2y^{-5}\cdot x^{-3}y\cdot \frac{1}{3}x^{-1}y^3\) into simpler terms \(x^{-2}\) and \(y^{-1}\), ultimately converting into positive exponents in the final answer.

Multiplication of Variables

Multiplying variables is a straightforward arithmetic operation, but it has unique aspects when variables have exponents. The product involves more than just multiplying the base numbers; the exponents must be handled according to specific rules.

When dealing with the multiplication of variables, follow these steps:

• Matching Bases: Ensure that you are only adding exponents for the same base. For instance, combining \(x^3\) with \(x^{-1}\) is valid, but you cannot combine \(x^3\) directly with \(y^3\).
• Summing Exponents: When multiplying terms with the same variable, add the exponents: \(x^a \times x^b = x^{a+b}\).

In our exercise example, you're interacting with multiple terms of \(x\) and \(y\), each with varying exponent values. By effectively adding these together, you'll reach simplified product expressions, which are much easier to work with in subsequent calculations.