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Chapter 1: Problem 7

Multiply. $$-14 \cdot(-3)$$

Short Answer

Expert verified
The product of \(-14\) and \(-3\) is \(42\), since both numbers are negative and the absolute values of them are \(14\) and \(3\), respectively. Thus, \(-14 \cdot(-3) = 42\).

Step by step solution

01

Multiply the absolute values of the numbers

First, we will find the absolute values of the two numbers, which are the numbers without their negative signs. The absolute value of -14 is 14, and the absolute value of -3 is 3. Now, we will multiply these absolute values together: \(14 \cdot 3 = 42\)
02

Determine the sign of the product

Since both numbers were negative, their product will be positive. As a general rule, if the signs of two numbers being multiplied are the same, the product will be positive.
03

Write the final answer

Now, we can write the final answer by keeping the positive sign from step 2 and using the product of the absolute values from step 1: \[-14 \cdot(-3) = 42\] So, the product of -14 and -3 is 42.

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Key Concepts

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

Understanding Absolute Value
Absolute value refers to the distance of a number from zero, without considering its direction on the number line. It's always a non-negative value.
  • The absolute value of -14 is 14.
  • The absolute value of -3 is 3.
To find the product of negative numbers, we can first use their absolute values, making calculations easier by temporarily ignoring the negatives. In our example, evaluate: \[14 \cdot 3 = 42\]This gives us the magnitude of the product without worrying about its sign just yet.
Exploring Integer Multiplication Rules
Integer multiplication follows specific rules that determine the sign of the product. These rules depend on whether the numbers are positive or negative:
  • Positive 脳 Positive = Positive
  • Negative 脳 Negative = Positive
  • Positive 脳 Negative = Negative
  • Negative 脳 Positive = Negative
Understanding these multiplication rules helps us decide whether the product will have a positive or negative sign. In our example, both numbers are negative, so\(-14 \cdot (-3)\)The product is positive because the rule states that multiplying two negatives results in a positive.
Multiplying Negative Numbers
Multiplying negative numbers might seem tricky, but it鈥檚 simple when you understand the concept. Negative numbers represent values below zero, often visualized on a number line.When two negative numbers are multiplied, think of it as reversing twice:
  • A negative number reverses direction once.
  • Another negative reversal turns it around again.
Thus, reversing direction twice takes you back to the original direction, resulting in a positive outcome. So when you multiply \(-14\) and \(-3\), you end up with a positive product.In summary, multiplying two negative numbers always yields a positive product, as seen with:\[-14 \cdot (-3) = 42\]

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Most popular questions from this chapter

Write a mathematical expression for each and simplify. The sum of \(-21\) and 13.

The 1999 Hennessey Viper Venom can go from 0 to 60 mph in 3.3 sec. The 2000 model goes from 0 to \(60 \mathrm{mph}\) in \(2.7 \mathrm{sec},\) which is a decrease of \(0.6 \mathrm{sec}\) to \(\mathrm{go}\) from 0 to \(60 \mathrm{mph} .\)

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Applications of Perimeter, Area, and Volume: Use 3.14 for \(\pi\) and include the correct units. Arun's Pet Supplies sells two medium-sized rectangular fish tanks. Model A has dimensions of \(30^{\prime \prime} \times 10^{\prime \prime} \times 14^{\prime \prime}\) while model \(B\) has dimensions of \(24^{\prime \prime} \times 13^{\prime \prime} \times 16^{\prime \prime}\) a) How many cubic inches of water will each tank hold? b) How many gallons of water will each tank hold? (Hint: Change cubic inches to cubic feet, then use the relationship \(1 \mathrm{ft}^{3}=7.48 \mathrm{gal}\))

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