Problem 44 - Elementary and Intermediate Algebra

Question

Use the quotient rule for exponents to simplify each expression. Write the results using exponents. $$ \frac{(0.25 y)^{9}}{(0.25 y)^{3}} $$

Step-by-Step Solution

Answer

\((0.25 y)^6\)

Step 1: Identify the Quotient Rule for Exponents

The quotient rule for exponents states that for any non-zero number \(a\) and integers \(m\) and \(n\), \(\frac{a^m}{a^n} = a^{m-n}\). This rule allows us to divide powers with the same base by subtracting the exponents.

Step 2: Apply the Quotient Rule

Given the expression \(\frac{(0.25 y)^9}{(0.25 y)^3}\), identify the base \(0.25y\) which is common to both the numerator and the denominator. Use the quotient rule: \(\frac{(0.25 y)^9}{(0.25 y)^3} = (0.25 y)^{9-3}\).

Step 3: Simplify the Exponents

Subtract the exponents: \(9 - 3 = 6\). Thus, \((0.25 y)^{9-3} = (0.25 y)^6\).

Step 4: Write the Final Simplified Expression

The expression \(\frac{(0.25 y)^9}{(0.25 y)^3}\) simplifies to \((0.25 y)^6\). This is the simplified expression using exponents.

Key Concepts

Simplifying Exponents

Exponents can initially seem complex, but the simplification rules help. One primary method is the quotient rule for exponents. This rule allows you to simplify expressions involving division of like bases raised to certain powers. Here's how it works: if you have an expression in the form of \( \frac{a^m}{a^n} \), where \( a eq 0 \), you can rewrite it as \( a^{m-n} \). It's like saying how many more times you multiply the base in the numerator compared to the denominator.
A step-by-step approach may help you visualize this:

• Identify the common base in both the numerator and denominator.
• Subtract the exponent in the denominator from the exponent in the numerator.
• Rewrite the expression using the result as the new exponent.

This rule simplifies expressions quickly and saves you from having to manually expand and simplify each base and power individually.

Algebraic Expressions

Algebraic expressions represent numbers and operations concisely. They include variables, numbers, and operations like addition or multiplication. To solve or simplify these expressions, clarity is key. Before you apply any arithmetic operation or rule, scrutinize each element.
When handling expressions with exponents, especially those seeking simplification, grasp what's represented:

• Variables, such as \( y \) in our example, can hold different values, and this impacts the overall expression.
• Numerical coefficients, like \( 0.25 \), directly influence the magnitude of the expression's outcome.
• Recognize expressions as blocks; this helps in applying operations or rules like the quotient rule effectively.

Simplicity forms when complex forms are broken down, making them easier to manipulate and understand.

Powers with the Same Base

Working with powers comes with its advantages, especially when all powers share the same base. Under these conditions, it becomes straightforward to apply consistent rules for operations like multiplication and division.
In our context, both \((0.25y)^9\) and \((0.25y)^3\) share the same base \(0.25y\). The operations involving these powers become easier when this is noted.

• When multiplying powers with the same base, add the exponents.
• When dividing powers with the same base, subtract the exponents (as per the quotient rule).
• Ensure that the base is correctly identified before applying any operation.

By acknowledging the same base, we minimize potential mistakes and enhance our ability to execute algebraic manipulations smoothly. This shared understanding forms the backbone for effective algebraic problem solving.