Problem 103 - Elementary Algebra

Question

For exercises 101-104, evaluate. Use a scientific or a graphing calculator. $$ 54-(-12)-9 \div(-3)^{2} $$

Step-by-Step Solution

Answer

65

Step 1 - Identify operations

The expression involves subtraction, division, and exponentiation: \[54 - (-12) - 9 \div (-3)^2\]

Step 2 - Resolve exponentiation

Calculate the exponentiation first: \[(-3)^2 = 9\]

Step 3 - Perform division

Substitute the result of the exponentiation and perform the division: \[9 \div 9 = 1\]

Step 4 - Simplify expression with addition and subtraction

Replace the obtained value and simplify the expression step-by-step: \[ 54 - (-12) - 1 \] This simplifies to: \[54 + 12 - 1\]

Step 5 - Perform addition and final subtraction

Complete the addition and subtraction: \[54 + 12 = 66\] followed by \[66 - 1 = 65\]

Key Concepts

Order of Operations

In algebra, the order in which you perform operations can change the result of your calculations. This is known as the 'order of operations,' often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right)).

It's important to follow these rules to ensure accuracy. In our exercise, we follow PEMDAS strictly:

• First, address any operations within parentheses.
• Second, resolve any exponents.
• Third, perform multiplication and division from left to right.
• Finally, do addition and subtraction from left to right.

Exponentiation

Exponentiation is a mathematical operation involving two numbers: the base and the exponent. In our exercise, \[ (-3)^2 \], -3 is the base and 2 is the exponent. When we raise a number to a power, we multiply it by itself as many times as the exponent indicates.

So, \[ (-3)^2 \] means:

\[ (-3) \times (-3) = 9 \]

This is crucial because incorrectly handling exponents can lead to wrong answers.

A common mistake is to ignore the negative sign or to incorrectly multiply, so be careful!

Division

Division splits things into equal parts. In this exercise, after computing the exponent, we perform division: \[ 9 \ (-3)^2 = 9 \ 9 = 1 \].

Always ensure you correctly replace the results back into your original expression. Here, we replaced the \(9 \ (-3)^2\) with 1.

Addition and Subtraction

For addition and subtraction, follow the left-to-right rule. Once we have simplified our expression: \[54 - (-12) - 1\], we must deal with our remaining operations.

First, handle the double negative, \54 - (-12)\ becoming \54 + 12\ = 66.

Finally, subtract the remaining 1: \66 - 1 = 65\

These steps complete our calculation.

Work step-by-step to keep track and avoid errors!