Problem 107 - Elementary and Intermediate Algebra

Question

Use the associative property of multiplication to find each product. $$ -0.2(-10 \cdot 3) $$

Step-by-Step Solution

Answer

The product is 6.

Step 1: Understand the Associative Property

The associative property of multiplication states that when three or more numbers are multiplied, the product is the same regardless of grouping. Mathematically, this is expressed as \((a \cdot b) \cdot c = a \cdot (b \cdot c)\).

Step 2: Reorganize the Expression

Given the expression \(-0.2(-10 \cdot 3)\), we can apply the associative property to regroup the multiplication terms. This becomes: \((-0.2 \cdot -10) \cdot 3\).

Step 3: Calculate First Pair

Now, calculate the product of \(-0.2\) and \(-10\). Multiplying two negative numbers gives a positive product: \(-0.2 \cdot -10 = 2\).

Step 4: Calculate the Final Product

With the partial product \(2\), multiply by \(3\): \(2 \cdot 3 = 6\).

Step 5: Conclusion

Thus, the product of the given expression \(-0.2(-10 \cdot 3)\) is \(6\).

Key Concepts

Multiplication

Multiplication is one of the fundamental operations in math, symbolized by the \(\cdot\) or \(\times\) sign. It involves combining equal groups to find a total. For example, if you have 3 groups of apples with 4 apples in each group, then in total, you have \(3 \times 4 = 12\) apples.
Multiplication is commutative, meaning the order of factors does not change the product (i.e., \(a \cdot b = b \cdot a\)). It also has an associative property which is very useful when dealing with several numbers. This property states that the way numbers are grouped in a multiplication problem does not change the product. For example, \((2 \cdot 3) \cdot 4 = 2 \cdot (3 \cdot 4)\).

• This property is particularly helpful when simplifying expressions.
• It allows you to rearrange terms to make calculations easier.
• This means you can group numbers together in a flexible way.

Understanding multiplication, especially its properties, lays the groundwork for tackling more complex algebraic expressions.

Negative Numbers

Negative numbers are numbers that are less than zero, shown with a minus sign (\(-\)). They can be a bit tricky, but they follow clear rules when used in multiplication and other operations. When you multiply two negative numbers together, the result is always a positive number. This seems a little strange at first, but it can be understood with a simple rule that negative times negative equals positive, illustrated as \((-a) \cdot (-b) = a \cdot b\). Conversely, if you multiply a negative number by a positive number, the result is negative. For instance, \(-3 \cdot 4 = -12\).

• This happens because multiplying by a negative is like "flipping" the sign of a number.
• If you have a negative number of times something good happens, on a larger scale, it's bad.
• Recognizing these patterns makes it easier to handle negative numbers in algebra.

So, when faced with calculations involving negatives, keep these rules in mind to avoid errors and get the correct product every time.

Algebra Concepts

Algebra involves using letters and symbols to represent numbers and quantities in formulas and equations. One of the basic principles in algebra is understanding and applying operations like multiplication in expressions and equations. Let's take a simple expression \(-0.2(-10 \cdot 3)\). By applying the associative property of multiplication, we can regroup the terms:

• Instead of multiplying \(-10 \cdot 3\) first, you can regroup to \((-0.2 \cdot -10) \cdot 3\).
• This lets you simplify parts of the expression step-by-step, making the calculation more manageable.
• The principles of multiplication and negative numbers also guide you to find the correct sign of the final result.

Algebra is like a toolbox with these rules and properties as your tools. The more you practice using them, the more intuitive solving problems can become. Remember that algebra is not just about finding the right answer but understanding the process and logic behind it.