Problem 114 - Elementary Algebra

Question

For the following problems, evaluate each numerical expression. $$ (-3)^{-2} $$

Step-by-Step Solution

Answer

Answer: $$\frac{1}{9}$$.

Step 1: Understand the expression

We have the numerical expression $$(-3)^{-2}$$ with a negative base (-3) and a negative exponent (-2).

Step 2: Use the rule for negative exponents

Remember that any number with a negative exponent can be represented as the reciprocal of that number with a positive exponent. Hence, we can rewrite the given expression: $$(-3)^{-2} = \frac{1}{(-3)^2}$$

Step 3: Evaluate the expression

Now we need to evaluate the expression inside the parenthesis and raise it to the power of 2: $$\frac{1}{(-3)^2} = \frac{1}{-3 \times -3} = \frac{1}{9}$$ So, the result of the numerical expression $$(-3)^{-2}$$ is $$\frac{1}{9}$$.

Key Concepts

Evaluating Numerical Expressions

When faced with numerical expressions such as \((-3)^{-2}\), it's important to approach the problem in a systematic way. The goal is to simplify the expression so we can understand its value.

First, identify the components of the expression: the base (-3) and the exponent (-2). Next, apply mathematical rules to evaluate the expression, which might include order of operations, handling of negative numbers, and exponentiation rules. In the given expression, the negative exponent indicates that we should take the reciprocal of the base raised to the positive equivalent of the exponent.

By systematically following the rules and understanding each step, students can demystify even the most initially perplexing expressions and find the correct evaluation with confidence.

Understanding the Reciprocal of a Number

The reciprocal of a number is simply the number turned upside down. In other words, for any non-zero number \(a\), its reciprocal is \(1/a\).

When dealing with negative exponents, knowing about reciprocals is crucial. The negative sign in an exponent is your cue to think 'reciprocal.' If you have \(a^{-n}\), this expression is equivalent to \(1/a^n\), which highlights the reciprocal relationship.

Application:

Our expression \((-3)^{-2}\) involves a negative exponent, so we take the reciprocal of (-3) squared. This transforms our initial complicated-looking expression into a simple fraction \(1/9\), which is the reciprocal of 9 or \(9^{-1}\). Understanding reciprocals can turn intimidating equations into easily solvable fractions.

Exponentiation Rules

Exponentiation rules are a set of guidelines that help us simplify expressions involving powers. One of these rules is the negative exponent rule, which states that a number raised to a negative exponent equals the reciprocal of that number raised to the corresponding positive exponent.

In applying this rule to \((-3)^{-2}\), we remember that a negative exponent doesn't affect the sign of the base; it only signifies that we should take the reciprocal. It's also important to point out that when raising a negative number to an even power, the result will always be positive (as a negative multiplied by a negative equals a positive). Therefore, \((-3)^2\) becomes 9, and \((-3)^{-2}\) simplifies to \(1/9\).

By understanding these rules, we can simplify and evaluate powers and roots effectively, allowing us to solve a wide range of mathematical problems efficiently and accurately.