Problem 8 - College Algebra

Question

Evaluate each exponential expression. $$ (-9)^{0} $$

Step-by-Step Solution

Answer

The value of (-9)^{0} is 1

Step 1: Understanding what the term implies

The given term is (-9)^{0}. Here, -9 is the base and 0 is the exponent

Step 2: Apply exponent rule

The exponent rule states that any non-zero number to the power of 0 is always 1. This is applicable even when the base is negative.

Step 3: Substitute and simplify

Using the rule, we substitute the given values into the rule to find that (-9)^{0} = 1

Key Concepts

Exponent Rules

Exponent rules are fundamental guidelines in mathematics that dictate how to handle expressions with exponents. An exponent refers to how many times a number, known as the base, is multiplied by itself. It is written as a small number above and to the right of the base number. There are several key rules:

• The Zero Exponent Rule: Any non-zero base raised to the zero power equals 1. For example, \((-9)^0 = 1\).
• Product of Powers Rule: To multiply two powers with the same base, add the exponents. For example, \(a^m \times a^n = a^{m+n}\).
• Power of a Power Rule: To raise a power to another power, multiply the exponents. For example, \((a^m)^n = a^{m\times n}\).

These rules are essential for simplifying complex expressions and solving equations. Understanding them is crucial for grasping more advanced mathematical concepts.

Evaluating Exponents

Evaluating exponents involves calculating the actual value of a number raised to a certain power. This process requires understanding how many times to multiply the base by itself based on the exponent number:

• A positive exponent indicates normal multiplication. For example, \(3^2\) means \(3 \times 3 = 9\).
• A zero exponent signifies that the result will be 1, regardless of what the base is, as long as the base is not zero. For instance, \((-9)^0 = 1\).
• A negative exponent denotes division and can be transformed into the reciprocal of the base raised to the opposite positive exponent. For example, \(2^{-3} = \frac{1}{2^3} = \frac{1}{8}\).

By familiarizing oneself with these points, evaluating complex expressions becomes straightforward and intuitive.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations that represent a particular value or set of values. They form the backbone of algebra and are used widely in various calculations. Key components include:

• Variables: Symbols (often letters) representing unknown values. For example, 'x' can represent any number.
• Constants: Numbers by themselves that have a fixed value, such as "5" in \(3x + 5\).
• Coefficients: Numbers multiplied by the variables. In \(3x\), '3' is the coefficient of 'x'.
• Operations: Include addition, subtraction, multiplication, division, and exponentiation.

Understanding how to manipulate algebraic expressions is crucial for solving equations and modeling real-life situations. While seemingly complex, mastering their properties can open the door to more advanced mathematics and applications.