Problem 44 - College Algebra with Modeling and Visualization

Question

Exercises 35-44: Use the product rule to simplify. $$ \left(3 x^{-4}\right)\left(2 x^{2}\right)\left(5 y^{4}\right)\left(y^{-3}\right) $$

Step-by-Step Solution

Answer

The simplified expression is \(30x^{-2}y\).

Step 1: Apply the Product Rule to x

To apply the product rule to the terms involving \(x\), identify the powers of \(x\) in the expression. You have \(x^{-4}\) from the first term and \(x^{2}\) from the second term. According to the product rule, multiply these by adding their exponents: \[x^{-4} imes x^{2} = x^{-4+2} = x^{-2}.\]

Step 2: Apply the Product Rule to y

Next, apply the product rule to the terms involving \(y\). Identify the powers of \(y\) in the expression. We have \(y^{4}\) and \(y^{-3}\). By the product rule, add the exponents:\[y^{4} imes y^{-3} = y^{4-3} = y^{1} = y.\]

Step 3: Combine the Constants

Multiply the constant coefficients from each term together. The constants from the terms are \(3\), \(2\), and \(5\). Multiply them:\[3 imes 2 imes 5 = 30.\]

Step 4: Write the Simplified Expression

Combine the results from Steps 1, 2, and 3:\[30 imes x^{-2} imes y = 30x^{-2}y.\] This is the simplified version of the original expression, using the product rule.

Key Concepts

Exponent Rules

Exponent rules are fundamental to simplifying expressions involving powers. These rules describe how to handle numbers and variables raised to exponents efficiently.

**Key Rules:**

• Product of Powers Rule: When multiplying two exponents with the same base, you add the powers. For example, if you have \(x^a \times x^b\), the result will be \(x^{a+b}\).
• Negative Exponent Rule: A negative exponent indicates a reciprocal. This means \(x^{-a} = \frac{1}{x^a}\).
• Zero Exponent Rule: Any non-zero base raised to the power of zero equals one, \(x^0 = 1\).

In our original problem, the product rule was used twice. First, it combined \(x^{-4}\) and \(x^{2}\), resulting in \(x^{-2}\). These steps help simplify the expression by consolidating terms with the same base.

Product Rule

The product rule is a straightforward and powerful rule applied in algebra to simplify expressions through multiplication. Its simplicity comes from handling exponents smartly.

In simple terms, the product rule allows you to multiply terms by just adding their exponents if their bases are the same. Here's how it works in our given expression:

• Look at the bases shared by terms, such as \(x\) and \(y\).
• Apply the rule by adding exponents: \(x^{-4} \times x^2 = x^{-4+2} = x^{-2}\).
• For \(y\), add: \(y^4 \times y^{-3} = y^{4-3} = y^1 = y\).

Using the product rule simplifies complex expressions into simpler forms, making them easier to understand and work with.

Polynomial Simplification

Polynomial simplification is the process of reducing expressions into their simplest form. It involves consolidating like terms and using algebraic rules.

The steps to simplifying a polynomial include:

• Combine Like Terms: Identify terms with the same variable and power, then combine them using the appropriate exponent rules.
• Multiply Constants: Calculate the product of constant coefficients in the expression. From the exercise, we multiplied \(3\), \(2\), and \(5\) to get \(30\).
• Reconstruct the Expression: Put your results together to form the simplified expression. From our example: \(30x^{-2}y\).

The goal is to present the polynomial in a compact and manageable form, making it easier to interpret and use in further calculations.