Problem 10 - College Algebra

Question

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

Step-by-Step Solution

Answer

The expression \((5-8)^0\) simplifies to 1.

Step 1: Understanding Exponents

Any number or expression raised to the power of zero is equal to one. This is a fundamental rule of exponents.

Step 2: Apply the Zero Exponent Rule

Given the expression \[(5 - 8)^0\].According to the zero exponent rule, \((a)^0 = 1\) for any non-zero value of \(a\). Therefore, \((5 - 8)^0 = 1\).

Key Concepts

Understanding Exponents

Exponents are a way to express how many times a number, known as the base, is multiplied by itself. For example, in the expression \(2^3\), the base is 2 and the exponent is 3. This means that the base, 2, is multiplied by itself three times: \(2 \times 2 \times 2 = 8\). Exponents allow us to write large numbers succinctly.

There are different properties of exponents that can help simplify expressions, such as the product of powers, quotient of powers, and power of a power, among others. For instance:

• Product of Powers: \(a^m \times a^n = a^{m+n}\)
• Quotient of Powers: \(a^m \div a^n = a^{m-n}\)
• Power of a Power: \((a^m)^n = a^{m \times n}\)

Understanding how to manipulate these properties is key to working with algebraic expressions, especially when simplifying them.

Exploring the Zero Exponent Rule

One of the most intriguing rules of exponents is the zero exponent rule. It states that any non-zero base raised to the exponent of zero is equal to one: \((a)^0 = 1\). This rule might seem a bit confusing at first, but it actually fits well with the patterns of exponents.

To understand why, consider the pattern of decreasing exponents:

• \(a^3 = a \times a \times a\)
• \(a^2 = a \times a\)
• \(a^1 = a\)
• \(a^0 = ?\)

If you notice the pattern, each step reduces the power by dividing by the base. Thus, when you reach \(a^0\), it becomes \(a^1 \div a = 1\).

This rule applies universally to any non-zero number or expression, making it a powerful tool in simplifying expressions.

Simplifying Algebraic Expressions

Expression simplification involves reducing an algebraic expression to its simplest form, making it easier to work with. Simplification can include combining like terms, using the distributive property, or applying rules of exponents. In the problem we considered, applying the zero exponent rule was crucial.

Let's break down the steps of simplification:

• Recognize and apply exponent rules: Use rules like the zero exponent rule to immediately reduce expressions with zero exponents.
• Combine like terms: Look for terms in the expression that have the same variables raised to the same power and add or subtract them accordingly.
• Use the distributive property: For expressions involving parentheses, distribute multiplication over addition or subtraction inside the parentheses where applicable.

By simplifying expressions, you not only find cleaner answers but also gain a better understanding of the relationships and interactions within the expression's parts.