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Chapter 1: Problem 55

For Problems 41-60, simplify each numerical expression. $$ \frac{4(-7)-8(-9)}{11} $$

Short Answer

Expert verified
The simplified expression is 4.

Step by step solution

01

Calculate the numerator part 1

First, evaluate the expression inside the numerator involving the first term by multiplying 4 by -7. \(4 imes (-7) = -28\)
02

Calculate the numerator part 2

Next, evaluate the second part of the numerator by multiplying -8 by -9. \((-8) imes (-9) = 72\)
03

Simplify the numerator

Sum the two results from Step 1 and Step 2 to get the complete numerator.\(-28 + 72 = 44\)
04

Perform the division

Now, divide the complete numerator by the denominator, 11.\(\frac{44}{11} = 4\)
05

Final Answer

The simplified numerical expression is 4.

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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's important to understand how the sign of the numbers affects the product. An integer can either be positive or negative.
  • If both integers have the same sign, the product is positive. For example, when multiplying a positive number like 3 by another positive number, the result is positive, for instance, \(3 \times 5 = 15\). The same goes for two negative numbers, such as \((-2) \times (-4) = 8\).
  • On the other hand, when integers have different signs,the product is negative. For example, multiplying\((-6) \times 2\) results in \(-12\).
Understanding these rules will help when simplifying expressions involving multiplication of integers. In the original exercise, multiplying 4 by -7 results in -28, showing a mix of signs as the reason the result is negative.
Division of Integers
Similar to multiplication, division of integers also follows certain rules related to the signs of the numbers:
  • If both the dividend and divisor have the same sign, the quotient is positive. For example,\((-10) \div (-5) = 2\), or \(6 \div 2 = 3\).
  • If the dividend and divisor have different signs, the quotient is negative. For instance,\((-20) \div 4 = -5\) and \(15 \div (-3) = -5\).
In the step-by-step solution, 44 (a positive number) is divided by 11 (alsoa positive number), resulting in a quotient of 4. This follows the samesign rule where two positive integers result in a positive quotient.
Order of Operations
The order of operations is a fundamental concept in mathematics that ensures consistent results when performing more than one operation in an expression. A powerful acronym to remember these rules is PEMDAS:
  • Parentheses first: Solve expressions inside parentheses.
  • Exponents (ie powers and roots, etc.).
  • Multiplication and Division: Solve from left to right.
  • Addition and Subtraction: Solve from left to right.
In the original exercise, the expression inside the fraction (numerator) is tackled first. This respects the sequence dictated by multiplication before addition, allowing the two multiplications to be computed prior to summing the results. This demonstrates a mindful application of the order of operations to achieve an accurate solution.

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Most popular questions from this chapter

State whether the expressions in each problem are equivalent and explain why or why not. \(9 m+8(3 p-q)\) and \(8(3 p-q)+9 m\)

Simplify each numerical expression. Don't forget to take advantage of the properties if they can be used to simplify the computation. $$17+(-18)-19-14+13-17$$

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. $$-3(x+2)-4(x-10)$$

State the property that justifies each statement. For example, \(3+(-4)=(-4)+3\) because of the commutative property for addition. $$-19+0=-19$$
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