Problem 35 - Algebra 1: Concepts and Skills

Question

Evaluate the expression without using a calculator. $$ \left(3^{2}\right)^{-1} $$

Step-by-Step Solution

Answer

The evaluated expression is \(1/9\).

Step 1: Simplify the Expression inside Parenthesis

The expression inside the parenthesis is \(3^2\), which simplifies to 9.

Step 2: Apply The Negative Exponent Rule

The negative exponent rule is \(a^{-n} = 1/a^n\). Applying this rule, \((9)^{-1}\) equals to \(1/9\).

Key Concepts

Negative Exponents

When you see a negative number as an exponent, don't worry, it's not as tricky as it may seem. The concept of negative exponents can be understood through a simple rule: to handle a negative exponent, you can flip the base to the denominator (or reciprocal) and turn the exponent positive. For example, if you have an expression like \( a^{-n} \), it equates to \( \frac{1}{a^n} \). This transformation moves the base \( a \) from the numerator to the denominator, turning the exponent positive in the process. This rule holds true for all except the base zero, as dividing by zero is undefined. Understanding this concept helps in simplifying expressions, especially when they involve fractions or decimal results. It’s a fundamental concept to grasp, aiding you in easier manipulation of algebraic expressions and solving equations effectively.

Simplifying Expressions

Simplifying expressions is an essential skill in algebra that makes solving problems more manageable. It involves reducing an expression into its simplest form by removing any redundant parts and applying mathematical rules. For instance, when you encounter an expression like \( 3^2 \), this means you are multiplying 3 by itself, resulting in 9. Putting it in its simplest form prevents lengthy equations from becoming too complicated. After simplifying inside any parentheses or brackets, always check if further simplification using exponent rules, such as distributing negative exponents, can be applied. This breaks down more complex equations into bite-sized steps, making them easier to handle.

Evaluating Expressions

Evaluating expressions is the process of finding the numerical value of an expression. It often involves combining the skills of simplifying and applying various mathematical rules. To evaluate an expression correctly, like \( (3^2)^{-1} \), follow a logical order. Start with operations inside parentheses, simplify them, and then apply any rules related to exponents or other operations. In our example, start by simplifying \( 3^2 \) to 9. Once simplified, apply the negative exponent rule, so \( 9^{-1} \) becomes \( \frac{1}{9} \). This step-by-step approach allows you to evaluate expressions systematically, ensuring you find accurate numerical values without a calculator. Practice these orderly simplifications to become proficient in quickly evaluating complex expressions.