Problem 73 - Elementary and Intermediate Algebra

Question

Evaluate each expression. $$ \frac{72-(2-2 \cdot 4)}{10^{2}-\left(9 \cdot 10+2^{2}\right)} $$

Step-by-Step Solution

Answer

The expression simplifies to 13.

Step 1: Resolve the expression inside the parentheses

First, evaluate the expression inside the parentheses in the numerator: \(2 - 2 \cdot 4\). According to the order of operations (PEMDAS/BODMAS), multiplication comes before subtraction. So,\(2 \cdot 4 = 8\).Substitute back:\(2 - 8 = -6\).Thus, the numerator becomes:\(72 - (-6)\), which is \(72 + 6 = 78\).

Step 2: Simplify the base of the denominator

Next, evaluate the expression inside the parentheses in the denominator:First, calculate \(9 \cdot 10\):\(9 \cdot 10 = 90\).Next, evaluate \(2^{2}\):\(2^{2} = 4\).Now substitute back into the expression:\(90 + 4 = 94\).

Step 3: Calculate the denominator

We need to finish calculating the denominator now:\(10^{2}\) which is \(100\), meaning the denominator becomes \[100 - 94 = 6\].

Step 4: Evaluate the entire expression

Replace the values in the corrected equation:\[\frac{78}{6}\].Finally, divide 78 by 6 to get the final answer:\(\frac{78}{6} = 13\).

Key Concepts

PEMDAS

Mathematics often involves a sequence of operations. To avoid confusion, mathematicians follow a rule to decide which operations to perform first: the order of operations. In the United States, this rule is often remembered with the acronym **PEMDAS**. Each letter in PEMDAS corresponds to:

• **P** stands for Parentheses: Solve expressions within parentheses first.
• **E** refers to Exponents: Calculate powers and square roots second.
• **MD** indicates Multiplication and Division: Perform these as they appear, from left to right.
• **AS** means Addition and Subtraction: Handle these last, also from left to right.

This method helps to ensure that everyone solves math problems in the same way, leading to consistent results.
For example, in the given expression \(2 - 2 \cdot 4\), according to PEMDAS, we multiply 2 and 4 first, before subtracting the result from 2. This avoids errors such as subtracting before the multiplication is keenly addressed.

BODMAS

While PEMDAS is a popular term in the United States, many other countries use **BODMAS** to convey the same order of operations in mathematics. BODMAS stands for:

• **B** is Brackets: Solve expressions within brackets first, akin to parentheses.
• **O** for Order: Tackle exponents and roots next, same as exponents in PEMDAS.
• **DM** is Division and Multiplication: Just like PEMDAS, handle these as they appear, from left to right.
• **AS** denotes Addition and Subtraction: Solve these operations last, in sequence from left to right.

The slight variation in acronym does not affect the underlying principle guiding the correct execution of calculations.
No matter if you use PEMDAS or BODMAS, both ensure clarity and consistency when calculating mathematical expressions.
In our example of the expression \((9 \cdot 10 + 2^{2})\), BODMAS confirms that multiplication and powers must be done before addition.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operations that define a mathematical phrase. Unlike simple arithmetic, algebra allows us to generalize operations and solve for unknowns. Components of algebraic expressions can include:

• **Numerals:** Like the number 72 in our example, which represents a constant quantity.
• **Variables:** Symbols that represent unspecified numbers or values.
• **Operators:** Include addition, subtraction, multiplication, division, and powers as seen in \(2 - 2 \cdot 4\).
• **Parentheses or Brackets:** Used to organize and dictate the priority in operations.

Algebraic expressions can be evaluated by following the order of operations to simplify them fully. As demonstrated, we begin by simplifying expressions within parentheses, followed by handling any exponents, and then proceed with multiplication or division operations, finishing with addition or subtraction.
This step-by-step approach was clearly illustrated in our task of simplifying the expression from the exercise, transitioning smoothly through its stages to reach a solution.