Problem 81 - Elementary Algebra

Question

For exercises 15-100, evaluate. $$ -6-3^{2}-(-4) $$

Step-by-Step Solution

Answer

-11

Step 1: Evaluate the exponent

First, evaluate the exponent in the expression. Calculate the value of \(3^{2}\): \(3^{2} = 3 \times 3 = 9\).

Step 2: Substitute the evaluated exponent back into the expression

Next, substitute the result of the exponentiation back into the original expression. This changes the expression to: \(-6 - 9 - (-4)\).

Step 3: Simplify the double negative

Simplify the double negative \(-(-4)\). A double negative becomes a positive, so \(-(-4) = +4\). The expression now is: \(-6 - 9 + 4\).

Step 4: Perform the subtraction

Combine the negative terms first. Start with \(-6 - 9\): \(-6 - 9 = -15\).

Step 5: Add the positive term

Finally, add the positive term to the result: \(-15 + 4\): \(-15 + 4 = -11\).

Key Concepts

Evaluating Expressions

Evaluating expressions means finding the value of an algebraic expression by following specific mathematical rules. To evaluate an expression, you need to substitute values for variables (if any) and then perform the mathematical operations in the correct order.
Let's look at an example. If the expression is \[-6-3^{2}-(-4)\], we don't have any variables to substitute. Evaluating this expression involves calculating each part step by step.
First, look for exponents, then handle any multiplications or divisions, and finally, tackle additions and subtractions. This systematic approach ensures you're adhering to the proper order of operations. The final evaluated expression will give you a single numerical value.

Exponents

Exponents are a way to show repeated multiplication of a number by itself. For example, \(3^{2}\) means you multiply 3 by itself: \(3 \times 3 = 9\).
In our example, the exponent operation \(3^2\) is evaluated first. This follows the order of operations rules, where we handle exponents before addition and subtraction.
Correctly evaluating exponents is crucial because it affects the rest of the calculations in the expression. Be careful with the base and the exponent to avoid mistakes.

Simplifying Expressions

Simplifying expressions makes a complex expression easier to work with by combining like terms and reducing it to its simplest form. In this exercise, simplifying involves handling negative signs and combining terms.
When simplifying \(-6 - 3^{2} - (-4)\), we dealt with the exponent first, then simplified any double negatives. For example, \(-(-4)\) becomes \(+4\), since two negatives give a positive.
Finally, you combine remaining numbers: \(-6 - 9 + 4\) simplifies to \(-11\). Breaking down expressions by simplifying steps prevents mistakes and helps in understanding each part clearly.

Order of Operations

The order of operations is a set of rules that dictate the sequence in which calculations should be performed to ensure accurate results. Commonly remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right)).
In our example expression \(-6 - 3^{2} - (-4)\), following the order of operations is essential. First, handle the exponent \(3^2 = 9\). Next, simplify the expression with double negatives: \(-(-4) = +4\).
Lastly, perform addition and subtraction from left to right: \(-6 - 9 + 4\). This ensures we get the correct final answer of \-11\. Without following PEMDAS, the calculations could be incorrect, leading to wrong results.