Problem 15 - Precalculus Mathematics for Calculus

Question

Evaluate each expression. (a) \(-3^{2}\) (b) \((-3)^{2}\) (c) \(\left(\frac{1}{3}\right)^{4}(-3)^{2}\)

Step-by-Step Solution

Answer

(a) -9; (b) 9; (c) \(\frac{1}{9}\).

Step 1: Evaluate \(-3^{2}\)

First, we need to understand that \(-3^{2}\) means only 3 is raised to the power of 2, then it is multiplied by -1. Therefore, calculate 3 squared to get: \(3^{2} = 9\), and then multiply by -1: \(-1 imes 9 = -9\). Thus, \(-3^{2} = -9\).

Step 2: Evaluate \((-3)^{2}\)

For \((-3)^{2}\), both the negative sign and 3 are raised to the power of 2. Calculate \((-3) \times (-3) = 9\). Thus, \((-3)^{2} = 9\).

Step 3: Evaluate \(\left(\frac{1}{3}\right)^{4}(-3)^{2}\)

First, calculate \(\left(\frac{1}{3}\right)^{4}\). This is \(\frac{1}{3} \times \frac{1}{3} \times \frac{1}{3} \times \frac{1}{3} = \frac{1}{81}\). Next, use the result from Step 2 for \((-3)^{2} = 9\). Multiply these: \(\frac{1}{81} \times 9 = \frac{9}{81} = \frac{1}{9}\). Thus, \(\left(\frac{1}{3}\right)^{4}(-3)^{2} = \frac{1}{9}\).

Key Concepts

Order of Operations

Mastering the order of operations is crucial for evaluating complex mathematical expressions accurately. The order of operations is a set of rules that dictates the sequence in which operations should be carried out to ensure that people arrive at the same result, regardless of the method they use. An easy way to remember the order is by using the acronym PEMDAS. This stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

• **Parentheses**: Always start by solving expressions inside parentheses.
• **Exponents**: Next, address any exponents in the expression.
• **Multiplication and Division**: Perform these from left to right.
• **Addition and Subtraction**: Finally, do these from left to right as well.

In the exercise, to correctly solve expressions like \(-3^{2}\) and \((-3)^{2}\), it is essential to determine what needs to be computed first, paying close attention to parentheses and exponents. Ignoring these rules can lead to errors, as seen in variations of expressions involving negative numbers and different operator orders.

Negative Numbers

Negative numbers often appear in mathematical expressions and can sometimes be confusing, especially when dealing with exponents. A key point to remember is that the placement of negative signs in relation to an exponent is critical.

For instance, in the expression \(-3^{2}\), the negative sign is not included in the exponentiation. This means only the 3 is squared, and not the entire term -3, hence it becomes \(-1 \times 3^2 = -9\).

On the other hand, when parentheses are used, as in \((-3)^{2}\), the negative sign is part of the base being squared. Thus, squaring the entire term \((-3)\) results in a positive outcome: \((-3) \times (-3) = 9\).

Understanding these differences ensures that you accurately solve expressions involving negative numbers and exponents, avoiding common pitfalls.

Fractional Exponents

Fractional exponents can initially seem daunting, but they follow straightforward principles if you break them down. A fractional exponent, such as \((\frac{1}{3})^{4}\), signifies that a number is raised to a power that is a fraction. Here, the expression implies that you multiply \(\frac{1}{3}\) by itself four times: \((\frac{1}{3}) \times (\frac{1}{3}) \times (\frac{1}{3}) \times (\frac{1}{3}) = \frac{1}{81}\).

Fractional exponents are a form of exponentiation that corresponds to roots. For example, an exponent of \(\frac{1}{n}\) signifies the nth root. They are very useful when dealing with transformations and scaling in mathematics.

The exercise combines such fractional calculations with integer exponents through multiplying two results: \((\frac{1}{3})^{4}\) and \((-3)^{2}\). Understanding how to manage each part separately and then together ensures clarity in solving these types of expressions.