Problem 10 - College Algebra with Corequisite Support

Question

For the following exercises, simplify the given expression. Write answers with positive exponents. \((5-8)^{0}\)

Step-by-Step Solution

Answer

The expression simplifies to 1.

Step 1: Understand the Zero Exponent Rule

The zero exponent rule states that any non-zero number raised to the power of zero equals one. This means that any expression that has an exponent of zero simplifies to 1, except for zero itself.

Step 2: Simplify the Expression Inside the Parentheses

The expression inside the parentheses is \(5 - 8\). We calculate this to get \(5 - 8 = -3\).

Step 3: Apply the Zero Exponent Rule

Now that we have simplified the expression inside the parentheses to \(-3\), we apply the zero exponent rule. \((-3)^0 = 1\). The base can be any number except zero, and with zero exponent, the result is always 1.

Key Concepts

Exponentiation

Exponentiation is a mathematical operation that involves two numbers: a base and an exponent. The base is the number being multiplied, while the exponent indicates how many times the base is multiplied by itself. For example, in the expression \(2^3\), the base is 2, and the exponent is 3, meaning 2 is multiplied by itself 3 times: \(2 \, \times \, 2 \, \times \, 2 = 8\).
For any base \(a eq 0\), if the exponent is zero, as in \(a^0\), the result is always 1. This is known as the Zero Exponent Rule, a key concept in exponentiation.
Understanding this rule helps when simplifying expressions such as those in this exercise, where a zero exponent significantly reduces complexity.

Simplifying Expressions

Simplifying expressions is about reducing a mathematical expression to its simplest form. This often involves performing arithmetic operations and applying algebraic rules to eliminate unnecessary components.
In the case of an expression like \((5 - 8)^0\), the first step is to simplify what's inside the parentheses. You do this by performing the subtraction: \(5 - 8 = -3\). Once the expression is simplified as much as possible, you can apply relevant mathematical rules such as the Zero Exponent Rule.
Simplifying expressions makes them easier to work with and helps reveal the fundamental components and relationships within, making problem-solving more straightforward.

Algebraic Rules

Algebraic rules are a set of guidelines that help manage and simplify the manipulation of mathematical expressions. Among these rules is the Zero Exponent Rule, which is crucial when dealing with expressions involving exponents.
This rule dictates that any non-zero number raised to zero results in 1. It's essential because it can simplify calculations dramatically, reducing complex equations to simpler numeric forms.
Other algebraic rules you might encounter include the Product of Powers Property and the Power of a Power Rule. These help manage expressions with multiple exponents and are vital tools for anyone working with algebraic expressions.