Question

You may use a graphing calculator to solve the following problems. True or False The number \(3^{-2}\) is negative. Justify your answer.

Short answer

False. The number \(3^{-2}\) is not negative because it equals \(\frac{1}{9}\), which is a positive fraction.

Step-by-step solution

Understand the problem

Our task is to determine if \(3^{-2}\) is negative or not and justify our answer. The number 3 is the base, and -2 is the exponent. The minus sign in the exponent is not saying that the base is negative.

Solve the exponent

Negative exponents follow this rule: \(n^{-m} = \frac{1}{n^{m}}\). So, \(3^{-2}\) can be rewritten as \(\frac{1}{3^{2}}\). When you calculate \(3^{2}\), you get 9. The answer to our problem should be \(\frac{1}{9}\).

Evaluate the answer

Now that we compute the value to be \(\frac{1}{9}\), we can see it's positive. Therefore, \(3^{-2}\) is not a negative number.

Key concepts

Understanding Exponentiation

Exponentiation is a mathematical operation, written as a base number raised to the power of an exponent. In simpler terms, when a number is raised to an exponent, it is multiplied by itself the number of times indicated by the exponent. For example, in the expression
\(2^3\), the number 2 is the base, and 3 is the exponent, signaling that 2 should be multiplied by itself 3 times, resulting in \(2 \times 2 \times 2 = 8\).

Negative Exponents

When it comes to negative exponents, the rules change slightly. A negative exponent tells us to take the reciprocal of the base raised to the absolute value of the exponent. Therefore, for any non-zero number 'n' and integer 'm', \(n^{-m} = \frac{1}{n^{m}}\). This means that no matter what the base is, as long as it's not zero, a negative exponent will result in a positive number because it represents the reciprocal of the base raised to a positive exponent. In conclusion, the expression \(3^{-2}\) results in a positive value of \(\frac{1}{9}\), not a negative.
It's essential to remember that negative exponents do not mean the number itself is negative; they merely modify how we approach the calculation.

Graphing Calculator Use

Graphing calculators are incredibly useful tools for visualizing mathematical concepts, including exponentiation. Students often turn to them to check their work or grasp complex ideas. Here’s how a graphing calculator can be used for understanding negative exponents:

• First, you would input the base number.
• Then, you'd include the exponentiation symbol, usually represented by a caret (^).
• Enter the negative exponent immediately after.
• Upon hitting the equals sign or similar command, the calculator displays the result.

Using a graphing calculator helps to confirm that a number with a negative exponent is indeed positive since the calculated result will be shown on the screen. For instance, if you input \(3^{-2}\), the calculator would display \(0.111...\), which represents the positive number \(\frac{1}{9}\). Graphing calculators can also visually represent functions involving exponents, allowing students to see how the function behaves and further confirming that negative exponents plot in the positive domain of a graph.
It is worth emphasizing the importance of familiarity with your calculator's functions and syntax to avoid errors and to correctly interpret the displayed results.

Numerical Expressions

Numerical expressions are combinations of numbers and operation symbols that, when evaluated, result in a number. In the context of negative exponents, the numerical expression involves using the exponentiation operation.
A considerable part of mastering numerical expressions is understanding the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Following this order is critical when evaluating expressions to reach the correct answer.

Simplified Numerical Expressions with Negative Exponents

When simplifying a numerical expression with a negative exponent, it's crucial to apply the negative exponent rule correctly to transition the expression into a more familiar form. The expression \(3^{-2}\) initially seems difficult because of the negative sign, but knowing the rule \(n^{-m} = \frac{1}{n^{m}}\) makes it simpler. By applying the rule, the expression simplifies to a recognizable positive fraction, making further evaluation or comparison straightforward.Understanding how to work with numerical expressions, including those with negative exponents, ensures you can tackle a wide range of mathematical problems. This solid foundation can be particularly beneficial when dealing with algebraic expressions where these rules are frequently applied.