Problem 104 - Elementary Algebra

Question

For exercises 101-104, evaluate. Use a scientific or a graphing calculator. $$ -8^{2}-4-15 \div(-3) $$

Step-by-Step Solution

Answer

-63

Step 1: Evaluate the exponent

First, evaluate the exponent part of the expression. Note that the expression \-8^{2} signifies -1 times 8 squared. Calculate 8 squared:\[ 8^2 = 64 \]Then, multiply by -1:\[ -8^2 = -64 \]

Step 2: Perform division

Next, perform the division part of the expression. Calculate \-15 \div (-3):\[ -15 \div (-3) = 5 \]

Step 3: Combine the results

Now, combine all parts of the expression:\[ -64 - 4 + 5 \]

Step 4: Perform addition and subtraction

Finally, perform the addition and subtraction operations from left to right:\[ -64 - 4 = -68 \]\[ -68 + 5 = -63 \]

Key Concepts

Order of Operations

When evaluating algebraic expressions, the order of operations is crucial. It determines which calculations to perform first to get the correct result.
Follow these steps:

• Parentheses: Any calculations inside parentheses come first.
• Exponents: Next, handle any exponents (or roots).
• Multiplication and Division: Then, proceed with multiplication and division from left to right.
• Addition and Subtraction: Lastly, finish with addition and subtraction from left to right.

In our exercise, we follow this order:
1. Exponents \(8^2\)
2. Division \(-15 \div (-3)\)
3. Addition/Subtraction \(-64 - 4 + 5\)
Ignoring the order can lead to incorrect answers.

Exponents

Exponents represent repeated multiplication of a number by itself. In our example, we have 8 squared, written as \(8^2\).
This means we multiply 8 by itself: \[ 8 \times 8 = 64 \]
However, since we are evaluating \(-8^2\), note the exponent only applies to 8, not the negative sign. Thus, we get:
\[ -8^2 = -64 \]
It's essential to carefully handle exponents, especially with negative signs. Misinterpreting this can often lead to wrong answers.

Division in Algebra

Division simplifies expressions by dividing one number by another. In our problem, we divide -15 by -3:
Evaluate: \[ -15 \div (-3) = 5 \]
Division rules for signs are:

• Positive \div Positive = Positive
• Negative \div Negative = Positive
• Positive \div Negative = Negative
• Negative \div Positive = Negative

Here, dividing two negatives results in a positive. Remember to follow these rules to avoid errors.

Addition and Subtraction

After handling exponents and division, we combine the remaining terms using addition and subtraction. In the final steps, we simplify:
\[ -64 - 4 + 5 \]
Start from the left and perform operations sequentially:
\[ -64 - 4 = -68 \]
Then, add the 5:
\[ -68 + 5 = -63 \]
This sequential approach minimizes mistakes when dealing with multiple operations in an expression.