Problem 147 - Elementary Algebra

Question

For the following problems, write each expression so that only positive exponents appear. Simplify \((-3)(-8)+4(-5)\)

Step-by-Step Solution

Answer

Question: Simplify the algebraic expression: \((-3)(-8) + 4(-5)\) Answer: 4

Step 1: Follow the Order of Operations

PEMDAS (Parentheses, Exponents, Multiplication and Division from left to right, and Addition and Subtraction from left to right) is the acronym that helps us to remember the order of operations in mathematics. According to PEMDAS rule, we should first perform multiplication, then addition.

Step 2: Perform Multiplication

We need to multiply the terms inside the brackets with their respective multipliers: \((-3)(-8) = 24\) \(4(-5) = -20\)

Step 3: Perform Addition

Now that we have simplified the multiplication part of the expression, we can proceed with addition: \(24 + (-20) = 24 - 20 = 4\) The simplified expression is 4.

Key Concepts

Positive Exponents

In mathematics, exponents tell us how many times to multiply a number by itself. A positive exponent means we perform this multiplication operation without negative signs.
For example, in the expression \( x^3 \), the 3 is a positive exponent, and it tells us to multiply \( x \) three times: \( x \times x \times x \).
In the exercise given, the focus is not on rewriting exponents but simplifying operations with integers and expressions ensuring all calculations yield positive results. Thus, maintaining only positive exponents is crucial when working with polynomials and real numbers to keep the values expressed in their simplest form.

Multiplication

Multiplication involves combining quantities. It is a fundamental operation and part of the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), which guides us through the order of operations.
In solving the expression \((-3)(-8)+4(-5)\), the multiplication is the first step after dealing with any exponents.
Here's how:

• Multiply \((-3)\) and \((-8)\), which gives \(24\) due to the rule that multiplying two negative numbers yields a positive result.
• Multiply \(4\) and \((-5)\) to get \(-20\). This product is negative because a positive times a negative results in a negative.

Understanding the signs during multiplication ensures you achieve the correct product, a key part of simplifying expressions.

Addition

Once multiplication is done, the next step involves simplifying results using addition. In mathematics, addition helps us find the total or sum of numbers or amounts.
After obtaining the products from the multiplication step:

• You have \(24\) from \((-3)(-8)\).
• And \(-20\) from \(4(-5)\).

Now, simply add these two results together: \(24 + (-20)\).
Subtract \(20\) from \(24\) (since adding a negative number involves subtraction) to get \(4\).
Having addition as a final step is crucial because it combines the results from earlier calculations, ensuring a simplified overall expression.

Simplifying Expressions

Simplifying an expression means breaking it down into its simplest form. This involves combining like terms and ensuring all operations follow the correct order, resulting in the minimal version of the expression.
In our problem,

• Start by performing the multiplication: \((-3)(-8) = 24\) and \(4(-5) = -20\).
• Then proceed to addition: Combine \(24\) and \(-20\) to yield \(4\).

Simplification helps not only in finding the numeric value but also boosts comprehension of the problem, making it easier to handle complex calculations in a structured format. By simplifying, we ensure the expression is easy to understand and utilize for further mathematical operations.