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Chapter 1: Problem 18

Multiply or divide as indicated. See Examples 4 and 5 . $$ -3 \cdot 8 $$

Short Answer

Expert verified
-24

Step by step solution

01

Identify the Numbers

We are given the expression \[-3 \cdot 8\].It involves two numbers: -3 and 8. The operation indicated is multiplication.
02

Multiply the Numbers

Perform the multiplication of the two numbers. Multiply the absolute values first, which gives you \[3 \times 8 = 24\].Since one of the numbers is negative, the result will also be negative.
03

Determine the Sign of the Product

The rule for multiplication dictates that a negative number multiplied by a positive number results in a negative product.Therefore, the result is\[-24\].

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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, represented with a minus sign, like
  • -1,
  • -3, or
  • -5.

They reflect a direction or frequency opposite to positive numbers. For example, a temperature of -3掳C is below freezing, while 3掳C is above.
When multiplying or dividing, the sign of the number significantly impacts the outcome. Here are some simple rules for negative numbers in multiplication:
  • Negative times Positive equals Negative: For example, l\(-3 imes 8 = -24\).
  • Negative times Negative equals Positive: For example, i\(-3 imes -2 = 6\).

These sign rules are essential to remember, as they determine the overall sign of the product or quotient.
Product of Numbers
In math, the product of numbers is what's obtained from multiplying them together. This operation combines numbers into a single value. For example, multiplying 3 and 4 gives a product of 12.
When dealing with multiplication, it's often demonstrated as:
  • The concept of commutativity: \(a \times b = b \times a\).
  • The use of associative property: \((a \times b) \times c = a \times (b \times c)\).
  • Multiplying any number by 1 leaves it unchanged, called the identity property: \(a \times 1 = a\).

In the example \(-3 \cdot 8\), you're looking for the product, which involves dealing with a negative and a positive number. Calculating products with different signs follows specific rules that dictate the final sign of the result.
Absolute Values
Absolute value refers to how far a number is from zero on the number line, without considering its direction. It's always a non-negative number. For instance, both 3 and -3 have an absolute value of 3.
Absolute values are vital because they allow us to focus solely on the size, not the direction or sign. This is helpful in multiplication, especially when considering the product of negative numbers.
  • The absolute value is denoted as \(|a|\) and considered like this:
    • If \(a\) is positive or zero, \(|a| = a\),
    • If \(a\) is negative, \(|a| = -a\).

In our step-by-step solution for \(-3 \cdot 8\), the absolute values are 3 and 8. Multiplying these gives 24. Only after this do we apply direction rules to determine that the result carries a negative sign, hence \(-24\).

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Most popular questions from this chapter

Use the distributive property to find the product. See Example II. $$ 0.5 x(6 y-3) $$

Evaluate each expression when \(x=9\) and \(y=-2 .\) See Example 12 $$ \frac{y^{2}+\sqrt{x+7}}{|3 x-y|} $$

Complete the statement to illustrate the given property. \(7 \cdot 1=\quad\)_____Multiplicative identity property.

Simplify each expression. See Examples \(11,14,\) and \(15 .\) $$ \frac{1}{4}(8 x-4)-\frac{1}{5}(20 x-6 y) $$

Complete the statement to illustrate the given property. \(9 y+(x+3 z)=\quad\)_____Associative property.
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