Problem 34 - Algebra and Trigonometry

Question

\(29-46\) Simplify each expression. $$ z^{5} z^{-3} z^{-4} $$

Step-by-Step Solution

Answer

The simplified form is \( z^{-2} \).

Step 1: Understand the Problem

The expression given is \( z^{5} z^{-3} z^{-4} \). We need to simplify this expression, which means combining the powers of \( z \) into a single term by using the rules of exponents.

Step 2: Apply the Product of Powers Property

The product of powers property states that \( a^m \cdot a^n = a^{m+n} \). This means we can add the exponents of \( z \) since the base \( z \) is the same. Thus, we have: \( z^{5} \cdot z^{-3} \cdot z^{-4} = z^{(5 + (-3) + (-4))} \).

Step 3: Simplify the Exponent

Add the exponents together: \( 5 + (-3) + (-4) = 5 - 3 - 4 = -2 \). Thus, the simplified exponent is \(-2\).

Step 4: Write the Final Simplified Expression

Replace the expression with the simplified single power: \( z^{-2} \). This is the simplified form of the expression.

Key Concepts

Simplifying Expressions

Simplifying expressions is an essential aspect of algebra, allowing us to break down and condense complex mathematical statements. In this context, "simplifying" means rewriting the expression in a more compact form while keeping its value unchanged. For the problem at hand, simplifying serves to reduce the expression with multiple exponents of the same base into one single expression. For example, consider the expression \( z^5 z^{-3} z^{-4} \). Here, we deal with multiple powers of the same base \( z \), and the goal is to combine these powers into one succinct term. To achieve this, we must understand and apply the appropriate exponent rules, ending with a simpler version of the original problem. Through simplification, mathematical equations become easier to handle and solve, paving the way for further calculations and insights.

Product of Powers Property

One of the fundamental rules to grasp when dealing with exponents is the Product of Powers Property. This property comes in handy, especially when multiplying expressions with the same base. The rule states that \( a^m \cdot a^n = a^{m+n} \), which means you just add the exponents when you multiply like bases.
Let's see how this applies in practice with the example \( z^5 z^{-3} z^{-4} \). Notice that the base, \( z \), is consistent across all terms. Therefore, by using the Product of Powers Property, you can add the exponents together:

• The first term has an exponent of 5, expressed as \( z^5 \).
• The second term has an exponent of -3, formatted as \( z^{-3} \).
• The third term an exponent of -4, written as \( z^{-4} \).

By adding these exponents (\( 5 + (-3) + (-4) \)), we can summarize all three parts as \( z^{-2} \). Recognizing and applying this rule helps in efficiently simplifying the expression.

Negative Exponents

Negative exponents can initially seem intimidating, but they represent something quite simple: reciprocal relationships. The rule for negative exponents tells us that a negative exponent means "take the reciprocal." For instance, \( z^{-2} \) translates to \( \frac{1}{z^2} \). This means that instead of multiplying \( z \) twice, you divide by \( z \) twice.
Understanding this helps in multiple scenarios:

• Negative exponents offer a method to convert between multiplication and division easily.
• They provide a way to express diminishment or decrease in multiplication terms.

In our simplified expression \( z^{-2} \), the negative sign indicates that rather than multiplying \( z \) by itself, the expression involves the reciprocal, leading to \( \frac{1}{z^2} \). Such insights assist in not only rewriting expressions but also in comprehending deeper mathematical concepts related to growth and decay.