Problem 14 - Algebra 1: Concepts and Skills

Question

Write the product in simplest form. $$\frac{y}{16} \cdot \frac{4 y^{4}}{y^{2}}$$

Step-by-Step Solution

Answer

The simplified form of the given expression is \( \frac{y^{3}}{4}\).

Step 1: Write down the problem

The given expression is \(\frac{y}{16} \cdot \frac{4 y^{4}}{y^{2}}\) and the goal is to simplify this.

Step 2: Apply the rule of the multiplication

The rule for multiplying fractions states that we multiply the numerators together to get the numerator of the product, and multiply the denominators together to get the denominator of the product. Applying this, we get \( \frac{(y \cdot 4y^{4})}{(16 \cdot y^{2})}\).

Step 3: Simplify the expression

Simplify to get \( \frac{4y^{5}}{16y^{2}}\).

Step 4: Simplify further by dividing both top and bottom to y squared

We can simplify this further by dividing both the numerator and denominator by \(y^{2}\). We are allowed to do this as long as \(y \neq 0\). This gives us \( \frac{4y^{3}}{16}\).

Step 5: Final simplification

The number 16 in the denominator can divide the number 4 in the numerator, giving us \(\frac{y^{3}}{4}\).

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and the operations of addition, subtraction, multiplication, and division. In the exercise, we've got an expression composed of fractions, each containing variables and coefficients. Here, the variable is \(y\), and it's combined with constants 16 and 4 and different powers. These expressions allow us to model real-world situations in algebraic terms.

When simplifying algebraic expressions, we aim to make them more manageable and easier to understand or work with. This involves reducing the expression to its simplest form by performing arithmetic operations and using algebraic rules. The goal of simplifying is to make the expression as compact as possible while retaining its original meaning. This process often includes steps such as factoring, distributing, and reducing like terms.

The Art of Multiplying Fractions

Multiplying fractions involves a straightforward rule that can make the process easy to remember and apply. When you multiply fractions, it’s important to multiply the numerators together and the denominators together. In the earlier step of the exercise, we saw this rule in action when we multiplied the top parts (numerators) and bottom parts (denominators) of the fractions.

Here’s a quick breakdown of how it works:

• Multiply the numerators: \( y \cdot 4y^4 = 4y^5 \).
• Multiply the denominators: \( 16 \cdot y^2 = 16y^2 \).

This method simplifies the process and sets the stage for further simplification. It’s important to follow this approach accurately to ensure you’re working with the correct numbers throughout the problem.

Decoding Exponents in Fractions

Exponents express repeated multiplication of a number or variable. In algebraic expressions, exponents follow specific rules that help simplify expressions efficiently. For instance, \( y^4 \) means \( y \cdot y \cdot y \cdot y \). When dealing with fractional expressions such as \( \frac{4y^4}{y^2} \), you can simplify by subtracting the exponents, using the quotient rule.

• Quotient Rule: \( \frac{y^m}{y^n} = y^{m-n} \). Apply this rule to get from \( \frac{4y^4}{y^2} \) to \( 4y^{4-2} = 4y^2 \).

Understanding how to manipulate exponents helps streamline the simplification process. By identifying opportunities to apply these rules, you can easily reduce complex expressions into simpler forms.

Mastering the Simplification Process

Simplification is the process of transforming a mathematical expression into its most reduced form. In this exercise, after applying the multiplication of fractions and knowing how to handle exponents, we moved to simplify the final expression. This involves several systematic steps:

• After multiplying, you simplify the fraction \( \frac{4y^5}{16y^2} \) by dividing both the numerator and denominator by \( y^2 \), resulting in \( \frac{4y^3}{16} \).
• The next step is reducing constants by finding the greatest common factor. Here, 4 divides 16 to result in the final simplified form: \( \frac{y^3}{4} \).

Always remember to check for common factors and simplify both the numeric and variable parts of the expression where possible. This structured approach ensures the expression is reduced correctly, leaving no further simplification possible.