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Chapter 10: Problem 65

Simplify. $$ 8(-7)+6(-5) $$

Short Answer

Expert verified
-86

Step by step solution

01

Multiply each term

Start by individually multiplying the constants in each term. First, multiply 8 and -7. Then, multiply 6 and -5.
02

Calculate the individual products

Calculate the product of 8 and -7: \[ 8 \times (-7) = -56 \]Then, calculate the product of 6 and -5: \[ 6 \times (-5) = -30 \]
03

Add the results

Now add the individual results together: \[ -56 + (-30) \]Since adding a negative is the same as subtracting, simplify it to: \[ -56 - 30 = -86 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Multiplication of Integers
When multiplying integers, it is crucial to understand the signs and their effects on the final result.
Multiplying two positive integers always results in a positive product, while multiplying two negative integers also results in a positive product.
However, if you multiply a positive integer by a negative integer, the result is always negative.
For our exercise, we had two such multiplications:
  • 8 and -7, where 8 is positive, and -7 is negative. The product is \(8 \times (-7) = -56\).
  • Similarly, for 6 and -5, since 6 is positive and -5 is negative, the product is \(6 \times (-5) = -30\).
Understanding these rules helps in performing multiplication of integers accurately.
Adding Negative Numbers
Adding negative numbers is essential to understanding in arithmetic.
When you add a negative number, it is similar to subtracting its absolute value.
For instance, in our exercise, we added -56 and -30:
  • The first step is to recognize this as \(-56 + (-30)\).
  • This addition is equivalent to subtracting 30 from -56.
  • Because we are combining the impacts of two negative numbers, the result becomes more negative.
Therefore, \(-56 + (-30) = -86\). Knowing this makes it easier to handle sums involving negative numbers.
Basic Arithmetic Operations
Basic arithmetic operations include addition, subtraction, multiplication, and division, which are fundamental in simplifying expressions.
In our example, we combined these operations:
  • We started with multiplication of integers: \(8 \times (-7)\) and \(6 \times (-5)\), which resulted in -56 and -30 respectively.
  • Then, we moved on to addition, which involved adding negative numbers: \(-56 + (-30) \).
  • Since our result was -86, recognizing how multiplication and addition work together in sequences is crucial.
Mastering these operations allows you to simplify and solve various mathematical expressions efficiently.

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