Problem 70 - Algebra 1: Concepts and Skills

Question

Evaluate the expression \(3 \cdot 3^{-5}\) A. \(\frac{1}{81}\) B. \(\frac{1}{5}\) C. 45 D. 81

Step-by-Step Solution

Answer

The correct answer is A. \(3 \cdot 3^{-5} = \frac{1}{81}\)

Step 1: Applying the rule of negative exponents

We rewrite \(3 \cdot 3^{-5}\) by changing the negative exponent to a positive exponent in the denominator. It now becomes \(3 \cdot (1/3^{5})\).

Step 2: Simplify the expression

Now, we simplify the expression within the parenthesis. \(3^5 = 243\) so now our expression is \(3 \cdot (1/243)\).

Step 3: Further Simplification

We simplify the multiplication by cancelling the 3 with the 243 in the denominator. This gives \(1/81\).

Key Concepts

Negative Exponents

The concept of negative exponents can be initially confusing, but it's essentially a way to express division as a multiplication by a reciprocal. Whenever you have an expression like \(a^{-n}\), it doesn't mean that the result is negative, but rather you would write the expression as \(\frac{1}{a^n}\). For example, \(3^{-5}\) translates to \(\frac{1}{3^5}\). In our particular exercise, this concept helps us transform \(3 \times 3^{-5}\) into a form that involves a simple fraction, thus simplifying the problem at hand.

To practice, consider \(2^{-3}\), which is the same as \(\frac{1}{2^3}\) or \(\frac{1}{8}\). With these simple transformations, you can handle negative exponents with ease.

Exponent Rules

Understanding exponent rules is crucial in simplifying algebraic expressions that include powers. These rules govern how to multiply, divide, and raise powers to powers when dealing with exponents. Some fundamental rules include:

• The Product Rule: \(a^m \times a^n = a^{m+n}\)
• The Quotient Rule: \(\frac{a^m}{a^n} = a^{m-n}\)
• The Power Rule: \((a^m)^n = a^{mn}\)

Applying these rules systematically can simplify complex expressions into more manageable forms. In the exercise, by understanding that we can multiply bases with the same value, we were able to simplify the expression effectively.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is a key skill in algebra that helps make complex problems more tractable. The goal is to transform the expression into its simplest form or minimum number of terms. This often involves:

• Combining like terms
• Applying exponent rules
• Doing arithmetic operations in the correct order

For our exercise, once we applied the negative exponent rule, it was a matter of simplification. We converted \(3^{-5}\) into \(1/3^5\) and then multiplied it times 3, which is part of simplification. The result was a basic fraction, \(1/81\), showcasing how these techniques can yield simple solutions.

Arithmetic Operations

Arithmetic operations are the foundation of all mathematical concepts and comprise addition, subtraction, multiplication, and division. Correctly executing these operations, and in the right order, is essential when evaluating expressions—particularly when they include exponents and variables. The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), guides us in the sequence. In the provided exercise, after using the exponent rules, we simplified the expression by multiplying, which is an arithmetic operation. We performed division within the multiplication process, cancelling out the common factor of 3. This is an elegant example of how arithmetic operations and algebraic concepts interplay to simplify and evaluate expressions.