Problem 31 - Algebra A Combined Function

Question

Perform each indicated operation. $$ 8^{2}+(-13)-4(-2) $$

Step-by-Step Solution

Answer

59

Step 1: Solve the Exponent

First, we calculate the exponent. We have the expression \(8^2\), which means multiply 8 by itself, i.e., \(8 \times 8 = 64\).

Step 2: Multiply and Subtract

Next, we handle the multiplication and subtraction in the remaining terms: \(-4(-2)\). The multiplication of two negative numbers results in a positive number: \(-4 \times -2 = 8\).

Step 3: Combine Results

Now, sum up all the results to get the final answer. Add the squared result and the product, then subtract 13: \(64 + 8 - 13\).

Step 4: Simplify

Perform the addition and subtraction: \(64 + 8 = 72\), then \(72 - 13 = 59\).

Key Concepts

Exponents

Exponents are a fundamental concept in mathematics, involving a number being raised to a power. The number being multiplied is known as the base, while the power it’s raised to is called the exponent. For example, in the expression \(8^2\), 8 is the base and 2 is the exponent. To solve it, multiply the base by itself as many times as dictated by the exponent. So, here it's \(8 \times 8\) which equals 64.
An easy way to remember this is:

• The base is how many times you multiply a number.
• The exponent tells you how many times to multiply it by itself.

Recognizing and solving exponents is crucial for correctly following the order of operations.

Multiplication with Negative Numbers

Multiplying negative numbers can sometimes be tricky, but it's all about remembering the rules. When multiplying two numbers, observe their signs as they determine the product's sign:

• Positive times Positive = Positive
• Negative times Negative = Positive
• Negative times Positive (and vice versa) = Negative

In our example, \(-4(-2)\), both numbers are negative. Following the rule, multiplying them gives a positive result, \(8\).
A helpful tip is to think of multiplying two negative numbers as flipping their signs back into positive. This logic keeps arithmetic systematic, making such operations easier to perform accurately.

Addition and Subtraction

The final step in solving the expression is to handle the addition and subtraction. To effectively manage this, remember the order of operations, typically known by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). It's key to tackle operations in the right sequence.
After addressing exponents and multiplying, sum up the intermediate results: \(64 + 8 - 13\). Start by adding \(64 + 8 = 72\). Next, subtract the remaining value: \(72 - 13 = 59\). Here are some quick tips:

• Complete additions before subtractions within a single step unless dictated otherwise by parentheses or other operations.
• Work sequentially from left to right for operations at the same level.

By adhering to the structured approach, you ensure accurate and effective calculation every time.