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Chapter 1: Problem 41

For Problems 41-60, simplify each numerical expression. $$ 3(-4)+5(-7) $$

Short Answer

Expert verified
-47

Step by step solution

01

Apply the Multiplication

First, calculate the products within the expression. Multiply 3 by -4, and 5 by -7 separately.- Multiplying 3 by -4: \[3(-4) = -12\]- Multiplying 5 by -7: \[5(-7) = -35\]
02

Add the Results

Now, take the two products from Step 1 and add them together.Calculate:\[-12 + (-35)\]- This is the same as subtracting 35 from -12:\[-12 - 35 = -47\]
03

Final Conclusion

The simplified form of the expression \(3(-4)+5(-7)\) is \(-47\).

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Key Concepts

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

Multiplication of Integers
Multiplication of integers involves multiplying whole numbers that can be positive or negative. Here's how this works:
  • When both integers have the same sign, the product is positive. For example, multiplying two positive integers like 3 and 4 gives 12, and multiplying two negative integers like -3 and -4 gives 12 as well.
  • When the integers have different signs, the product is negative. In the given problem, you multiply 3 (positive) by -4 (negative), resulting in -12. Similarly, 5 (positive) multiplied by -7 (negative) results in -35.
Remember, the absolute value of the product is always the product of the absolute values of the numbers. The sign is determined by the rules above.
Understanding these simple rules will help you tackle multiplication of integers with confidence in any mathematical expression.
Addition of Negative Numbers
Adding negative numbers can be a bit tricky at first, but it's straightforward once you get the hang of it.
  • When you add two negative numbers, you're essentially adding their absolute values together and keeping the negative sign. For example, the expression \(-6 + (-2)\) can be thought of as taking 6 away and then 2 more, totaling -8.
  • In our problem, when you add -12 and -35, you add the absolute values of 12 and 35, giving you 47, and then because both numbers are negative, you keep the negative sign, resulting in -47.
The key takeaway is to view the addition of negative numbers as cumulative subtraction.
Order of Operations
The order of operations is a critical concept in simplifying expressions like the one presented in this exercise. It ensures that everyone performs calculations in the same sequence.

Here's a simple way to remember the order of operations:
  • Parentheses
  • Exponents (powers and roots)
  • Multiplication and Division (from left to right)
  • Addition and Subtraction (from left to right)
Often remembered by the acronym PEMDAS, this order ensures consistency and clarity in solving math problems.
In our given expression, we apply these rules by first carrying out the multiplications \(3(-4)\) and \(5(-7)\) before performing the addition of the results. Sticking to this order gives us the accurate, simplified form of the expression. By practicing these steps, you'll find simplifying expressions becomes much easier.

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Most popular questions from this chapter

Simplify each algebraic expression by combining similar terms. $$9(x+4)-(x-8)$$

State the property that justifies each statement. For example, \(3+(-4)=(-4)+3\) because of the commutative property for addition. $$-119+119=0$$

Simplify each algebraic expression by combining similar terms. $$8 x+9+14 x-3 x-14$$

Your friend keeps getting an answer of 45 when simplifying \(3+2(9)\). What mistake is he making and how would you help him?

State the property that justifies each statement. For example, \(3+(-4)=(-4)+3\) because of the commutative property for addition. $$[5(-8)] 4=5[-8(4)]$$
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