Problem 73 - Elementary Algebra

Question

For exercises 1-80, evaluate. $$ [5(19-10)-3]-(8-4)^{2} $$

Step-by-Step Solution

Answer

The expression evaluates to 26.

Step 1: Simplify inside the parentheses

Start by simplifying the expressions inside the parentheses. First, calculate the expressions within the parentheses. 1. Calculate a. \(19 - 10 = 9\) b. \(8 - 4 = 4\)

Step 2: Perform operations inside the brackets

Next, use the results from Step 1. Substitute back into the expression inside the brackets and perform the calculations. 1. Calculate a. \(5 \times 9 - 3 = 45 - 3 = 42\)

Step 3: Handle the exponent

Calculate the exponent part of the expression: 1. Calculate a. \(4^2 = 16\)

Step 4: Simplify the expression

Finally, substitute back and perform the subtraction: 1. Substitute and subtract a. \[42 - 16 = 26\]

Key Concepts

Order of Operations

One of the most important rules in math is the order of operations. This tells us the sequence in which we should evaluate an expression. A common way to remember the order is the acronym PEMDAS:

• Parentheses
• Exponents (or powers)
• Multiplication
• Division
• Addition
• Subtraction

Following PEMDAS helps ensure that everyone gets the same answer when solving an expression. If we ignore this order, we might end up with incorrect results. For example, in our problem \[ [5(19-10) - 3] - (8-4)^2 \], following the order of operations is crucial to solving it correctly. We start with the parentheses, then handle the exponent, and finally, perform subtraction.

Parentheses

Parentheses (also known as brackets) are used to group parts of an expression that should be calculated first. Whenever we see parentheses, we need to evaluate the expression inside them before moving on to other operations. In the given problem, the first step was to simplify inside the parentheses:

1. For \( 19 - 10 \), we get \ 9.
2. For \( 8 - 4 \), we get \ 4.

After simplifying these, the expression inside the brackets becomes simpler. This step is necessary to make sure we respect the order of operations.

Exponents

Exponents represent repeated multiplication of the same number. In our problem, we see the expression \( (8-4)^2 \). After simplifying the parentheses to \ 4 \, we need to calculate the exponent.

This means \ 4 \ raised to the power of \ 2 \, which is \( 4 \times 4 = 16 \). We handle exponents after the parentheses have been simplified and before multiplication or division. This step is essential as it converts powers into a simpler number for further operations.

Subtraction

Subtraction is the process of taking one number away from another. In the given problem, subtraction appears multiple times. To correctly solve the expression, we should follow the order of operations and perform subtraction at the appropriate step:

1. First, subtract the numbers inside the parentheses: \( 19 - 10 \) becomes \ 9 \, and \( 8 - 4 \) becomes \ 4.
2. Then, after dealing with multiplication and exponents, we end up with \ 42 - 16.

This final subtraction gives \ 26. Subtraction must be done after any parentheses, exponents, and multiplicative steps, according to PEMDAS.

Multiplication

Multiplication is one of the core arithmetic operations, which means adding a number to itself a certain number of times. In the given problem, multiplication occurs within the brackets:

\[ 5 \times (19 - 10) - 3 \]

After simplifying inside the parentheses, it becomes \ 5 \times 9 - 3 \. We first perform the multiplication \( 5 \times 9 = 45 \) and then handle the subtraction to get \ 42. Multiplication must be done before subtraction (unless parentheses dictate otherwise). This ensures each operation is performed in the correct order, yielding the correct result.