Question

Evaluate the expression. Write fractional answers in simplest form.\(3^{0}\)

Short answer

The answer is 1.

Step-by-step solution

Identify the Base and the Exponent

In the expression \(3^{0}\), 3 is the base and 0 is the exponent. The problem is asking us to calculate the value of this expression.

Apply the Rule of Exponents

The rule of exponents states that any non-zero number raised to the power of 0 is always equal to 1. Thus, \(3^{0}\) is equal to 1.

Key concepts

Rules of Exponents

Understanding the rules of exponents is fundamental in algebra. These rules allow us to perform operations involving powers more efficiently and correctly. When we talk about exponents, we're referring to the number that indicates how many times the base, a certain number, is multiplied by itself. For example, in the expression 3^4, the base is 3 and the exponent is 4, meaning 3 is multiplied by itself 4 times (3 * 3 * 3 * 3).

One of the most essential rules is the one used in the exercise: any non-zero base raised to the exponent of zero equals one. This rule is ubiquitous, and it simplifies what could be a complex calculation into a simple understanding that, regardless of the base (as long as it's not zero), the answer is always 1. This rule is represented mathematically as a^0 = 1 for any a ≠ 0.

Other important rules include the product of powers rule (a^m * a^n = a^(m+n)), the quotient of powers rule (a^m / a^n = a^(m-n)), and the power of powers rule ((a^m)^n = a^(m*n)). These form a foundation that allows for the simplification and manipulation of exponentiated terms.

Simplifying Expressions

Simplifying expressions is a procedure in algebra that makes them easier to understand and work with. Simplification might involve condensing an expression by combining like terms, utilizing the rules of exponents, or factoring. The goal is to rewrite the expression in its least complicated form without changing its value.

For instance, when we simplify the expression 3^0 using the rule that any number raised to the power of zero is one, we are applying the simplification process to obtain the number 1, which is the simplest form of the expression. In more complex cases, simplifying may include multiple steps such as distributing multiplication over addition or subtraction, combining similar terms, and reducing fractions to their simplest form. These steps help to unveil the core components of an expression, making it clearer and usually shorter. Simplification is particularly useful when solving equations, as it can make the solutions more apparent.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operation symbols that represent a specific mathematical relationship or quantity. These expressions can include terms, which are the separate parts of the expression combined by addition or subtraction, and factors, which are the numbers, variables, or parenthetical expressions multiplied together within a term.

In the given problem, 3^0 is a simple algebraic expression with a base of 3 and an exponent of 0. It illustrates the idea that algebraic expressions can be subject to rules such as the rules of exponents. By understanding the underlying concepts, such as the properties of zero as an exponent, students can evaluate and simplify even more complex algebraic expressions. The beauty of algebra comes from the way these expressions can model real-world problems and the manipulations that we can perform on them to derive meaningful solutions.