Problem 64 - College Algebra with Modeling and Visualization

Question

Exercises \(55-64:\) Use the power rules to simplify the expression. Use positive exponents to write your answer. $$ \left(\frac{2 x y}{3 z^{5}}\right)^{-1} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{3z^5}{2xy} \).

Step 1: Understand Negative Exponent Rule

The negative exponent rule states that \( a^{-n} = \frac{1}{a^n} \). This means that taking the reciprocal of a number with a negative exponent will change the sign of the exponent.

Step 2: Apply Negative Exponent Rule to the Expression

Apply the negative exponent rule to the expression \( \left(\frac{2xy}{3z^5}\right)^{-1} \). This results in swapping the numerator and the denominator of the fraction and changing the exponent to positive: \( \frac{3z^5}{2xy} \).

Step 3: Simplify the Fraction

The fraction is already in its simplest form: \( \frac{3z^5}{2xy} \). Ensure that the exponents are positive and there are no common factors between the numerator and the denominator.

Key Concepts

Exponent Rules

Algebraic expressions often involve exponents, which signify how many times a number, called the base, is multiplied by itself. Understanding exponent rules is crucial to simplify these expressions. One of the key exponent rules is when multiplying two powers that have the same base, you can add their exponents. For example, \[ a^m \times a^n = a^{m+n} \].
Another important rule is the power of a power rule, guiding us that when an exponent is raised to another exponent, you multiply the exponents. This is shown as:\[ (a^m)^n = a^{m\times n} \].
Using these rules correctly allows you to manipulate and simplify complex algebraic expression, aiding greatly in achieving the correct solutions efficiently.

Negative Exponent

The concept of a negative exponent might seem confusing at first, yet it is straightforward once you grasp the principle it follows. In simple terms, a negative exponent indicates a reciprocal. For instance, when you see \( a^{-n} \), it is transformed to \( \frac{1}{a^n} \).
The key thing to remember is: \( a^{-n} \) does not represent a negative number but instead a fraction. In our example, the expression \( \left( \frac{2xy}{3z^5} \right)^{-1} \) flips, turning into \( \frac{3z^5}{2xy} \) because of the negative exponent rule.
This rule is crucial because it simplifies expressions by eliminating negative exponents, thus making further calculations easier.

Fraction Simplification

Fraction simplification is an essential skill in algebra, especially when dealing with expressions involving variables and exponents. The goal is to reduce the fraction as much as possible while keeping it mathematically equivalent to the original fraction.
To simplify a fraction:

• Cancel any common factors shared between the numerator and the denominator.
• Ensure all exponents are positive.
• Simplify any complex expressions within the fraction.

In the example \( \frac{3z^5}{2xy} \), the fraction is already simplified as there are no common factors to cancel between the numerator and the denominator.
Also, make sure to check the exponents, ensuring that they are positive and the expression is in its most reduced form. Mastery of this allows you to handle more complicated expressions with ease.