Problem 31 - Elementary and Intermediate Algebra

Question

Simplify. $$ 8 \cdot 7+6 \cdot 5 $$

Step-by-Step Solution

Answer

The simplified expression is 86.

Step 1 - Perform the Multiplications

First, perform the multiplications in the given expression. Compute both products separately:\[ 8 \times 7 = 56 \ 6 \times 5 = 30 \]

Step 2 - Sum the Results

After calculating the products, add the resulting values together:\[ 56 + 30 \]

Step 3 - Simplify the Sum

Finally, perform the addition to simplify the expression fully:\[ 56 + 30 = 86 \]

Key Concepts

Order of Operations

Understanding the order of operations is crucial when simplifying algebraic expressions. The order of operations, also known as PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)), helps ensure that we perform calculations in the correct sequence. In this exercise, there are no parentheses or exponents, so we start with multiplication before moving on to addition. Writing out such expressions clearly, as shown in the steps, helps us avoid common mistakes and confusion.
Always tackle the multiplication steps before you proceed to addition.

Multiplication

In our expression, we had two multiplication operations:

• 8 multiplied by 7
• 6 multiplied by 5

To simplify, we perform these operations first:
\[ 8 \times 7 = 56 \]
\[ 6 \times 5 = 30 \]
After calculating these products, we are left with two numbers to sum up. Multiplication breaks down larger problems into more manageable steps. This avoids adding complexities with mixed operations.

Addition

Once the multiplication steps are complete, we move to addition. At this stage, our expression simplifies to:
A straightforward addition:
\[ 56 + 30 = 86 \]
Adding these two numbers gives us the final answer. Addition consolidates all the values into a final simplified number. Breaking down each step and verifying your intermediate results help ensure the final answer is correct.
Use addition to bring together the products obtained through the preceding multiplication steps efficiently.