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Chapter 1: Problem 46

For Problems 41-60, simplify each numerical expression. $$ (-7)(-6)+(-4)(-3) $$

Short Answer

Expert verified
The simplified expression is 54.

Step by step solution

01

Multiply the First Pair of Negative Numbers

First, multiply the two negative numbers in the first pair: \[ (-7) \times (-6) \]. Multiplying two negative numbers results in a positive number, so:\[ -7 \times -6 = 42 \].
02

Multiply the Second Pair of Negative Numbers

Next, multiply the two negative numbers in the second pair: \[ (-4) \times (-3) \]. Similarly, multiplying two negative numbers results in a positive number, so:\[ -4 \times -3 = 12 \].
03

Add the Results of the Two Multiplications

Finally, add the two results from the previous steps:\[ 42 + 12 \]. Adding these positive numbers results in:\[ 42 + 12 = 54 \].

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Key Concepts

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

Negative Numbers
Negative numbers are numbers less than zero, often found on the opposite side of a number line than positive numbers. Understanding their multiplication is crucial in algebra. A unique rule applies when you multiply two negative numbers: the product is a positive number. This is because multiplying negative numbers counteracts the negativity each one contributes, resulting in a positive outcome.

An easy way to remember this is to think of the signs. Multiplying a negative by a negative flips the sign, leading to a positive. For example:
  • \((-7) \times (-6) = 42\)
  • \((-4) \times (-3) = 12\)
By practicing various combinations, you can become more comfortable with this rule. Always remember, a negative times a negative equals a positive.
Multiplication
Multiplication is one of the core operations in mathematics, allowing us to add a number to itself a certain number of times. When dealing with negative numbers within multiplication, it鈥檚 crucial to know how signs affect the result. Here are some basic rules:
  • Positive multiplied by positive gives a positive product.
  • Negative multiplied by negative results in a positive product.
  • Positive multiplied by negative, and vice versa, yields a negative product.
When multiplying large numbers, it's helpful to break them into easier parts or use multiplication tables for quick reference. In the exercise, we simplified expressions like \((-7) \times (-6)\) and \((-4) \times (-3)\), both yielding positive results due to their negative factors.
Accurate multiplication is vital for simplifying more complex algebraic expressions, as each step builds on the previous one.
Numerical Expression Simplification
Simplifying numerical expressions is about breaking down complex mathematical statements into their simplest form. This is important in algebra, where expressions can get long and complicated.

The exercise showed us how to handle such expressions, using multiplication and addition:
  • First, the pairs of negative numbers were multiplied to turn them into positive products.
  • Next, these results were added together for the final simplified expression.
In this case,
  • \(42 + 12 = 54\)
Simplification not only makes expressions easier to handle but is also essential in checking your work for accuracy. Attention to detail in each step ensures that the final result reflects correctly simplified terms.

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Most popular questions from this chapter

Insert parentheses so that \(36+12 \div 3+3+6 \cdot 2\) is equal to \(55 .\)

Simplify each algebraic expression and then evaluate the resulting expression for the given values of the variables. \(5(x-y)-9(x+y)\) for \(x=4\) and \(y=-4\)

Simplify each algebraic expression and then evaluate the resulting expression for the given values of the variables. \((a+b)-(a-b)\) for \(a=-16\) and \(b=14\)

Simplify each algebraic expression by combining similar terms. $$(y+3)-(y-2)-(y+6)-7(y-1)$$

State in your own words the associative property for addition of integers.
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