Problem 87 - College Algebra with Modeling and Visualization

Question

Exercises \(65-90:\) Use rules of exponents to simplify the expression. Use positive exponents to write your answer. $$ \frac{4 x^{-2} y^{3}}{\left(2 x^{-1} y\right)^{2}} $$

Step-by-Step Solution

Answer

The simplified expression is \( y \).

Step 1: Simplify the Denominator

First, simplify the denominator \( \left( 2x^{-1}y \right)^2 \). This involves squaring both the coefficients and the variables within the parenthesis.- \( (2)^2 = 4 \) - \( (x^{-1})^2 = x^{-2} \) - \( (y)^2 = y^2 \) Therefore, the denominator becomes \( 4x^{-2}y^2 \).

Step 2: Set Up the Expression for Simplification

Now substitute the simplified denominator into the expression. The expression becomes:\[ \frac{4x^{-2}y^3}{4x^{-2}y^2} \]

Step 3: Simplify the Expression by Cancelling Terms

Cancel the common terms in the numerator and the denominator. First, note that \( 4 \) in the numerator and \( 4 \) in the denominator cancel each other out. Then, simplify the exponents:- \( x^{-2} \) in the numerator and denominator cancel out completely.- For \( y \), apply the exponent rule \( y^{a}/y^{b} = y^{a-b} \): - \( y^{3-2} = y^{1} = y \)Thus, the simplified expression is \( y \).

Step 4: Write the Final Expression with Positive Exponents

The expression has already been simplified to use positive exponents. Thus, the final expression remains:\[ y \]

Key Concepts

Exponent Rules

Exponent rules are like the magic rules that help make complex expressions look simpler. When dealing with an expression that includes exponents, these rules guide you through the simplification process. Here are a few essential exponent rules that you must keep in mind:

• Product of Powers Rule: When multiplying two powers with the same base, simply add their exponents. For example, \( x^a \times x^b = x^{a+b} \).
• Power of a Power Rule: When raising a power to another power, multiply the exponents. For instance, \( (x^a)^b = x^{a \times b} \).
• Quotient of Powers Rule: When dividing two powers with the same base, subtract the exponents. This is shown as \( x^a / x^b = x^{a-b} \).
• Negative Exponent Rule: A negative exponent means you take the reciprocal of the base. For instance, \( x^{-a} = 1/x^a \).

Applying these rules helps you transform and simplify expressions to make them easier to work with.

Simplifying Expressions

Simplifying expressions is about making them easier to understand and work with. In our exercise, we encountered a fraction with exponents. The process of simplification involved the following:

• Simplify the Denominator: Start by dealing with what's inside the parentheses and apply the power of a power rule. For instance, when faced with \( (2x^{-1}y)^2 \), you square both the coefficient and the variables separately.
• Cancel Common Terms: After substituting the simplified denominator back into the expression, you can cancel out any common factors. For example, the factor \( 4 \) was present in both the numerator and denominator and was canceled out.

By following these steps, we can break down complex expressions into more manageable terms. This helps us not only solve problems efficiently but also understand the structure of the expression better.

Positive Exponents

Writing expressions with positive exponents is important for clarity and convention. Positive exponents are expressions where the exponent is greater than or equal to zero. Here's why and how we ensure our final expression uses positive exponents:

• Clarity: Positive exponents make expressions easier to read and compare. They're generally more intuitive, since they reflect how many times to multiply the base by itself.
• Conversion: If you encounter a negative exponent during simplification, convert it to a positive one by taking the reciprocal of the base. For example, if you have \( x^{-2} \), rewrite it as \( 1/x^2 \) to make it a positive exponent.

In our solved example, the final result, \( y \), perfectly exemplifies using positive exponents. Initially negative exponents in the process were canceled out to yield this simplified and clear form of the expression.