Problem 5 - Algebra and Trigonometry

Question

Evaluate each expression. $$ (-6)^{0} $$

Step-by-Step Solution

Answer

The expression evaluates to 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 one. This means that regardless of the value of the base, as long as it is not zero, the expression will always equal one.

Step 2: Apply the Zero Exponent Rule

In the expression \((-6)^0\), the base is \(-6\), which is a non-zero number. According to the zero exponent rule, raising \(-6\) to the power of zero results in 1.

Key Concepts

Exponentiation

Exponentiation is a mathematical operation involving two numbers: the base and the exponent. It represents the base multiplied by itself a certain number of times, indicated by the exponent. For instance, in the expression \(a^b\), \(a\) is the base, and \(b\) is the exponent. If \(b\) is a positive integer, such as 3, the expression translates into multiplying \(a\) by itself three times (\(a \times a \times a\)). Exponents are a shorthand method for writing repeated multiplication.

One specific rule within exponentiation is the zero exponent rule, which states that any non-zero number raised to the zero power equals one. This rule simplifies expressions like \((x^0)\) to 1, regardless of the base value, assuming it is not zero. Understanding these basic principles of exponentiation is crucial for evaluating expressions effectively.

Evaluating Expressions

The process of evaluating expressions involves simplifying them to find numerical values. When given an expression such as \((-6)^0\), we need to apply the rules of mathematics to simplify it. In this instance, the zero exponent rule is key.

By applying the zero exponent rule, we can transform \((-6)^0\) into 1. This is because, as the rule dictates, any non-zero number raised to the power of zero simplifies directly to one. By understanding the rules associated with exponents, evaluating expressions becomes more straightforward. Therefore, even when faced with a potentially complex-looking expression, applying these foundational rules allows us to simplify them efficiently.

Algebra Basics

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating these symbols. It underpins much of higher mathematics and provides a basis for understanding other mathematical concepts.

In algebra, understanding rules like the zero exponent rule is essential. These rules simplify expressions and help solve equations. Algebra involves operations such as addition, subtraction, multiplication, division, and exponentiation. Each operation obeys specific laws that, when learned, make algebraic expressions easier to handle.

For example, recognizing that any non-zero base raised to the zero power results in one simplifies our calculations. This basic knowledge reduces the complexity of algebraic expressions, ensuring they are not overly mystifying. Mastering these basic algebraic principles prepares students for more advanced topics by building a strong mathematical foundation.