Problem 25 - College Algebra and Calculus: An Applied Approach

Question

Simplify the expression.\((-5 z)^{3}\)

Step-by-Step Solution

Answer

\(-125z^3\)

Step 1: Apply product-of-powers rule

First, let's apply the product-of-powers rule to \((-5z)^3\), which gives \((-5)^3 * (z)^3\). This adheres to the rule where \((ab)^n = a^n * b^n\).

Step 2: Simplify the coefficients and variables separately

Solving for \((-5)^3\), which equals -125 and \(z^3\), the final expression becomes \(-125z^3\).

Step 3: Solution

The original expression simplifies as \(-125z^3\).

Key Concepts

Product of Powers Rule

The product of powers rule helps us simplify expressions where exponents are involved. It comes into play when we have a base with an exponent multiplied by the same base with another exponent. The rule states that you can add the exponents:

• If you have \(a^m \times a^n\), it becomes \(a^{m+n}\).
• This allows for simplification and makes complex expressions much easier to handle.

In the provided exercise \((-5z)^3\), the expression \((-5z)^3\) can be broken down using the product-of-powers rule to \((-5)^3 \times z^3\). This can be seen as an example of how the rule works effectively.
By applying this rule, it simplifies handling exponents, especially when dealing with multiple factors raised to the same power. Understanding this concept is key to mastering algebraic expressions in general.

Simplifying Expressions

Simplifying expressions is an essential skill in algebra. It involves reducing an expression to its simplest form. A simplified expression is easier to understand and work with. Here's how you can approach this:

• Break down complex parts using basic math operations and rules.
• Combine like terms and perform the operations across the expression.

In our example, once the product-of-powers rule was applied, we move to simplify further. We calculate \((-5)^3\), resulting in -125, and keep \(z^3\) as it is.
This way, the expression becomes -125z^3, which is the simplest form of the original expression. Simplification not only makes expressions more manageable but also prepares them for solving equations or further operations.

Algebraic Expressions

Algebraic expressions are combinations of numbers, letters, and arithmetic operations. They represent mathematical ideas in a concise form. In expressions, letters, typically referred to as variables, stand in for numbers. Here's what you need to know:

• An algebraic expression may contain constants (specific numbers), variables (symbols that imitate numbers), and operators (like +, -, *, /).
• These expressions follow a specific set of rules and operations.

In the given exercise, \((-5z)^3\) is an algebraic expression.
We see how algebraic expressions can incorporate exponents, making it necessary to apply rules like the product-of-powers to simplify them. Understanding this lets you manipulate the expressions to solve equations or model real-world phenomena. Getting comfortable with expressions and simplification techniques is fundamental to success in algebra and higher mathematics.