Problem 11 - Algebra and Trigonometry

Question

\(7-28\) Evaluate each expression. $$ (-6)^{0} $$

Step-by-Step Solution

Answer

The expression \((-6)^0\) evaluates to 1.

Step 1: Understanding the Zero Exponent Rule

When a number, except zero, is raised to the power of zero, the result is always 1. This is because any number to the power of zero represents an empty product, which is conventionally equal to 1.

Step 2: Apply the Zero Exponent Rule

The expression \[ (-6)^0 \] can be rewritten using the zero exponent rule as \[ 1 \] because any non-zero number to the power of zero is equal to 1.

Key Concepts

Exponentiation

Exponentiation is a fundamental concept in mathematics that extends the idea of repeated multiplication. If you have a number known as the base, raising it to an exponent tells you how many times to multiply the base by itself. For example, * Raised to the power of 3, it means multiplying 2 three times: \( 2^3 = 2 \times 2 \times 2 = 8 \).Exponentiation is not limited to whole numbers or positive exponents. Negative and fractional exponents, as well as zero exponents, all have their own rules. These rules exist to ensure that exponentiation works consistently across all numbers. A special rule worth noting in terms of exponentiation is the zero exponent rule, which states any non-zero number raised to the power of zero is always 1.

Powers of Numbers

The term "powers of numbers" refers to the result you get when a number is raised to a particular exponent. These powers have consistent and easy-to-remember patterns: * Raising any number to the first power leaves the number unchanged, like \( 5^1 = 5 \). * Raising any number to the second power means squaring it, which is multiplying it by itself, such as \( 7^2 = 7 \times 7 = 49 \). * When a number is raised to the zero power (as long as it is not zero), the result is 1. This is because the zero exponent represents an empty product, which by definition is 1. So, \( (-6)^0 = 1 \).This idea of raising numbers to different powers forms the building blocks for many algebraic operations, such as polynomial expressions and complex equations.

Algebra Basics

Algebra is the branch of mathematics dealing with symbols and the rules for manipulating those symbols. It provides the foundation for solving equations and understanding mathematical relationships. Algebra basics include working with terms, equations, and expressions. * **Terms** are individual numbers, variables, or the product of numbers and variables. For example, in the expression \( 3x + 2 \), both \( 3x \) and \( 2 \) are terms. * **Equations** are statements that two expressions are equal. They are the mathematical sentences formed by relating two expressions with an equal sign, as in \( 4x = 12 \). * **Expressions** are combinations of terms without an equal sign. For example, \( x^2 - 7x + 10 \) is an expression.Understanding exponentiation and powers of numbers is crucial in algebra becausemany algebraic expressions and functions involve exponents. The zero exponent rule is a part of these algebraic basics, helping simplify and evaluate expressions. For instance, knowing that \( (-6)^0 \) equals \( 1 \) allows students to simplify complex expressions easily. Together, these basics equip students to tackle more challenging mathematical tasks.