Problem 58 - Introductory and Intermediate Algebra for College Students 4th

Question

Use the order of operations to simplify each expression. $$\frac{-5(7-2)-3(4-7)}{-13-(-5)}$$

Step-by-Step Solution

Answer

The simplified expression is 2.

Step 1: Simplify expressions in parentheses

Simplify the expressions within the parentheses. The given expression modifies to \(\frac{-5(5)-3(-3)}{-13-(-5)}\).

Step 2: Perform multiplication with the results from step 1

Multiplication is the next operation to perform according to PEMDAS. The expression now becomes \(\frac{-25-(-9)}{-13-(-5)}\).

Step 3: Simplification

Simplify the numerator and denominator separately. This will make the expression \(\frac{-25+9}{-13+5}=\frac{-16}{-8}\).

Step 4: Division

Finally, by performing the division operation you get 2 as the result.

Key Concepts

Simplifying Expressions

Simplifying expressions is an essential skill that helps make complex mathematical problems more manageable. It involves reducing expressions to their most concise form, often by performing arithmetic operations such as addition, subtraction, multiplication, and division.
To simplify an expression like the given exercise, you often start by addressing operations inside parentheses or brackets. This is crucial because parentheses indicate which operations should be performed first. Once the expressions in parentheses are simplified, you can proceed to the next steps, which could include further simplification by combining like terms or reducing fractions.

• Start with any operations inside parentheses.
• Combine like terms.
• Use the order of operations to process arithmetic operations.

Simplifying expressions correctly ensures that calculations are accurate and efficient, which is vital in solving mathematical problems.

PEMDAS

PEMDAS is an acronym used to remember the order of operations: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). This order is crucial for solving mathematical expressions correctly.
In the provided problem, following PEMDAS ensures each operation is performed in the correct sequence:

• Parentheses: Simplify what's inside first.
• Multiplication and Division: Perform from left to right as they appear.
• Addition and Subtraction: Execute these last, also from left to right.

By following this order, you prevent common mistakes that can arise from performing operations at random.

Multiplication

Multiplication is one of the four basic arithmetic operations, symbolized by \(\times\) or \(\cdot\). In the context of the exercise, multiplication is performed after simplifying expressions inside parentheses.
Let's consider the step where multiplication occurs in the given expression:

• The expressions \(-5(5)\) and \(-3(-3)\) involve multiplication.
• Compute \(-5 \times 5 = -25\) and \(-3 \times -3 = 9\).

Multiplication can significantly change the values in an expression, especially when negative numbers are involved, as they can convert subtraction into addition due to negative signs.

Division

Division is another fundamental arithmetic operation, represented by the symbol \(\div\) or as a fraction. In the process of simplifying expressions, division often serves to reduce the final result to a single value.
In the problem, the division appears in the final step once both the numerator and the denominator have been simplified:

• The expression \(\frac{-25 + 9}{-13 + 5}\) leads to \(\frac{-16}{-8}\) after simplification.
• Divide \(-16\) by \(-8\) to get \(2\).

Division of two negative numbers results in a positive quotient, which is crucial for arriving at the correct final answer. This operation ensures the conversion of a complex expression into a simpler, actionable result.