Problem 4 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. $$ 3^{-1} $$

Step-by-Step Solution

Answer

\(\frac{1}{3}\)

Step 1: Understanding the expression

The expression given is \(3^{-1}\). Here, 3 is the base and -1 is the exponent. This falls under the rule of negative exponents.

Step 2: Applying the negative exponent rule

According to the negative exponent rule, \(a^{-n} = \frac{1}{a^n}\). Applying this rule to \(3^{-1}\), we get \(\frac{1}{3^1}\).

Step 3: Evaluating the expression

Since any number raised to the power of 1 is the number itself, \(\frac{1}{3^1}\) simplifies to \(\frac{1}{3}\).

Key Concepts

Evaluating Expressions

Evaluating expressions, especially those that include negative exponents, is a key skill in algebra that can initially seem a bit daunting. To evaluate the expression \(3^{-1}\), we must first understand each part of the expression. The number 3 is the base, while -1 is the exponent. Negative exponents can sometimes trick students into thinking they result in negative numbers, but in reality, they denote reciprocal values. This means converting the base into a fraction over 1. For \(3^{-1}\), the evaluation process involves applying a specific rule of exponents, which we'll delve into next. After understanding and applying the relevant rules, we simplify the expression to reach the answer.

Rules of Exponents

The rules of exponents are essential guidelines that help simplify expressions and perform calculations efficiently. These rules remove complexity and make it easier to work with expressions involving powers.

• Negative Exponent Rule: This rule states that \(a^{-n} = \frac{1}{a^n}\). Therefore, a negative exponent indicates the reciprocal of the base raised to the positive of that exponent.


• Zero Exponent Rule: Any non-zero base raised to the zero power is always 1, that is, \(a^0 = 1\).


• Positive Exponent Rule: This is where the base is multiplied by itself as many times as indicated by the exponent, i.e., \(a^n = a\times a\times a\ldots (n \text{ times})\).

When you apply these rules to \(3^{-1}\), using the negative exponent rule, we get \(\frac{1}{3^1}\). From here, simplify further to achieve the evaluated result of \(\frac{1}{3}\).

Exponents and Powers

Exponents and powers are foundational mathematical concepts used to express how many times a number, known as the base, is multiplied by itself. Understanding these concepts can make calculations straightforward. The base raised to an exponent is collectively referred to as a 'power'. For example, in \(3^4\), 3 is the base, and the exponent 4 signifies that 3 is multiplied by itself four times: \(3 \times 3 \times 3 \times 3\).However, when dealing with negative exponents, it's important to remember they transform the base into its reciprocal form; they don't indicate subtraction or negative numbers. With \(3^{-1}\), the power denotes that we take the reciprocal of 3 raised to the power of one, simplifying to \(\frac{1}{3}\). Having a strong grasp on these concepts ensures you can tackle any expression involving exponents and powers with confidence.