Problem 12 - Algebra 2 and Trigonometry

Question

In \(11-22,\) find the value of each expression when \(x \neq 0\) $$ (-5)^{0} $$

Step-by-Step Solution

Answer

The value of \((-5)^0\) is 1.

Step 1: Understanding the Zero Exponent Rule

The zero exponent rule states that any non-zero number raised to the power of zero is equal to 1. Mathematically, this is written as \(a^0 = 1\) if \(a eq 0\).

Step 2: Identify the Base and Apply the Rule

For the given expression \((-5)^0\), identify that \(-5\) is the base. Since \(-5\) is a non-zero number, apply the zero exponent rule to simplify \((-5)^0\) to 1.

Step 3: Check Conditions

Ensure that all conditions of the zero exponent rule are satisfied, namely that the base \(-5\) is indeed non-zero. Since this condition holds true, the expression simplifies correctly.

Key Concepts

Zero Exponent Rule

The Zero Exponent Rule is a fundamental principle in algebra that simplifies expressions involving exponents. According to this rule, any non-zero number raised to the power of zero is equal to 1. This can seem confusing at first, but it is a powerful tool for simplifying algebraic expressions. For example, \(a^0 = 1\) as long as \(a eq 0\).
The logic behind this rule comes from the properties of exponents. When we divide powers with the same base, we subtract the exponents: \(a^m/a^m = a^{m-m} = a^0\). Since any number divided by itself is 1, it follows that \(a^0 = 1\) if \(a eq 0\).
This rule only applies if the base is non-zero. If the base were zero, you'd have an indeterminate form. Thus, it is critical to ensure the base is non-zero when applying the Zero Exponent Rule.

Simplifying Expressions

Simplifying expressions is a key skill in algebra. It involves reducing an expression to its simplest form. This makes it easier to work with, especially when solving equations or performing other operations.
The process usually involves:

• Combining like terms
• Applying the rules of exponents
• Eliminating any unnecessary parts of the expression

In the context of exponents, to simplify an expression like \((-5)^0\), you apply the Zero Exponent Rule to get 1.
This process helps not just in computing answers quickly but also in understanding the behavior of expressions. Being familiar with these rules, including exponent rules, is crucial for more complex algebraic problems.

Exponentiation

Exponentiation is the process of raising a number, called the base, to a power, which is expressed as an exponent. In algebra, it's a shortcut for repeated multiplication. For example, \(3^4\) means \(3\) multiplied by itself \(4\) times: \(3 \times 3 \times 3 \times 3\).
The exponent indicates how many times the base is used as a factor. This operation is foundational in mathematics and frequently appears in formulas, equations, and real-world applications, such as calculating compound interest or exponential growth.
Understanding how to manipulate exponentiation, like using the Zero Exponent Rule, is essential for simplifying expressions and solving problems efficiently.

Non-zero Base Condition

The non-zero base condition is crucial when working with exponents, especially when applying the Zero Exponent Rule. This condition states that when using rules like \(a^0 = 1\), the base \(a\) must not be zero.
This is because raising zero to the power of zero, or \(0^0\), is considered an indeterminate form. Indeterminate forms do not have a specific value, and the rules for zero and other numbers don't apply the same way.
Therefore, whenever you're simplifying expressions with exponents and applying rules, always ensure the base isn't zero unless specific conditions permit—it maintains the accuracy and validity of your results.