Problem 68 - Elementary Algebra

Question

For exercises 15-100, evaluate. $$ -3^{2}-17(-1) $$

Step-by-Step Solution

Answer

-3^{2}-17(-1) evaluates to 8.

Step 1 - Evaluate the exponent

First, evaluate the exponent in the expression evaluate the exponent in the expression -3^{2}3^{2}equals 9 and we must now include the negation: -9 .

Step 2 - Multiply -17 with -1

Now, multiply -17 with -1. This gives: (-1) (-17) = 17.

Step 2 - Add the results

Now, add the results from steps 1 and 2: -9 + 17 We find that : equals=8.]

Key Concepts

Understanding Exponents

An exponent indicates how many times a number, called the base, is multiplied by itself. For instance, in the expression \(3^2\), 3 is the base and 2 is the exponent. Here, 3 is multiplied by itself: 3 * 3 = 9.

When evaluating expressions with exponents, it is essential to follow the order of operations (PEMDAS/BODMAS), where you first solve expressions inside parentheses, then exponents, followed by multiplication and division, and finally addition and subtraction.

For example, in the expression \(-3^2\), we evaluate the exponent first, \((3^2)= 9\), then apply the negation (more on this later), leading to -9.

The Role of Negation

A negation in mathematics refers to changing the sign of a number. For example, the negation of 9 is -9.

When you see a negative sign in front of an expression with an exponent, like \(-3^2\), it is crucial to apply the exponent first and then the negation: \(3^2 = 9\) and \(-9\).

Confusion often arises with placement of parentheses. For instance, \(-3^2\) is different from \((-3)^2\). The latter means you first take -3 and then square it, giving you \((-3) * (-3) = 9\). Always pay attention to whether the negative sign is inside or outside the parentheses.

Basics of Multiplication

Multiplication is one of the fundamental arithmetic operations where you combine groups of equal size. For instance, 3 multiplied by 4 (3 * 4) gives you 12.

When dealing with negative numbers, remember these important rules:

• A negative number multiplied by a positive number yields a negative result (e.g., -2 * 3 = -6).
• A negative number multiplied by another negative number results in a positive number (e.g., -2 * -3 = 6).

In our example, we need to multiply \(-17 * -1\). Applying our multiplication rules, the result is 17.

Performing Addition

Addition involves combining numbers to get a sum. In our example, we need to add -9 and 17. When adding a negative number to a positive number:

• If the positive number is larger, subtract the absolute values and take the sign of the larger absolute value (e.g., -9 + 17 = 8).
• If the negative number is larger, do the same but take the negative sign (e.g., -17 + 9 = -8).

Therefore, \(-9 + 17 = 8\), because the positive number 17 is larger than 9.

Remember to follow the rules for combining positive and negative numbers to avoid mistakes.